Essays on Stochastic Volatility

Essays on Stochastic V olatility

and

Random -Field M odels in Finance

H e l e n T s o u l o u v i

Thesis submitted in partial fulfilment of the requirements for the degree of

Doctor of Philosophy in Economics

LO N D O N SCHOOL OF ECONOM ICS

2005

UMI Number: U213592

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2

A bstract

In this thesis we develop random-field models for the implied volatility of equity options and the term structure of interest rates. Following a brief introduction to the topics of this thesis in chapter 1, chapter 2 models the Black-Scholes implied volatility of plain-vanilla European stock options as a random field with three parameters: current time, the maturity date and the exercise price of the corresponding option. In this model all plain-vanilla European options are needed to complete the market. Illiquid and exotic derivatives can be priced as a function of the stock price and the implied volatility surface.

In chapter 3 we develop a random-field model for forward interest rates with stochastic volatility. It is assumed that the forward rate volatility function can be decomposed into a deterministic function of the time to maturity and a maturity- independent stochastic process driven by a standard Brownian motion. The separability of the forward-rate volatility function allows closed-form solutions to be obtained for the prices of a number of interest rate derivatives: bond options, interest rate caplets, and interest rate spread options. Forward LIBOR and swap rates are modelled in a similar way, and closed-form solutions are derived for the prices of LIBOR caplets and swaptions.

In chapter 4 we estimate three random-field models of the term structure of interest rates: one model with deterministic forward-rate volatility, and two with stochastic forward-rate volatility. The models are estimated using seven years of daily UK and US forward rate data, spanning times to maturity between zero and 120 months. The parameters of each model are obtained by maximizing the likelihood function. We develop an importance sampling technique that substantially reduces the variance of the Monte Carlo estimator of the likelihood function in the case of stochastic volatility.

Acknow ledgem ents I wish to thank Antonio Mele, Angelos Dassios, and Vassilis Hajivassiliou for helpful comments and suggestions, and my parents for their support and encouragement.

3

Contents

List of Tables 5

1 Introduction 6

2 A random-field model for implied volatility 9

2.1 Introduction.............................................................................................. 9

2.2 Stochastic implied volatility m o d e l....................................................... 15

2.3 Pricing volatility derivatives........................................................................... 23

2.4 Pricing and hedging contingent c la im s......................................................... 29

2.5 Conclusion ..................................................................................................... 31

3 Random-field models of the term structure of interest rates with sto

chastic volatility 32

3.1 Introduction.............................................................................................. 32

3.2 A random-field model for the forward rate with stochastic volatility . . . 36

3.2.1 The model under the risk-neutral measure......................................... 36

3.2.2 Pricing bond options............................................................................ 39

3.2.3 Interest rate c a p le ts ............................................................................ 41

3.2.4 Interest rate spread op tions ................................................................ 43

3.2.5 The model under the physical measure.............................................. 45

3.3 Random-field models for LIBOR and swap rates with stochastic volatility 46

3.3.1 Introduction ........................................................................................ 46

3.3.2 LIBOR model........................................................................................ 47

3.3.3 Swap rate m o d e l.................................................................................. 50

3.4 Conclusion ..................................................................................................... 52

4 Estimation of random-field models of the forward rate with stochastic

volatility 53

4.1 Introduction......................................................................................................

4.2 The m odels ......................................................................................................

4.3 Estimation of the models ..............................................................................

4.3.1 Model I ................................................................................................

4.3.2 Model I I ................................................................................................

4.3.3 Model I I I .............................................................................................

4.4 The data .........................................................................................................

4.5 Empirical re su lts .............................................................................................

4.6 Conclusion ......................................................................................................

5 Conclusion

Bibliography

A The Ito-Venttsel formula

B Proof of lemma 2.3

C Proof of proposition 3.1

D Proof of proposition 3.2

E The forward-rate drift under the risk-neutral measure in section 4.2

F Graphs of exponential functions with estimated parameters in models

I, II, and III

53

57

61

61

63

67

70

71

74

76

78

88

88

89

90

90

92

G Monte Carlo study results 95

5

List o f Tables

2.1 Volatility diffusion equation p aram eters ..................................................... 11

4.1 Volatility function specification.................................................................... 58

4.2 Model I parameter estimates ....................................................................... 71

4.3 Model II parameter estim ates....................................................................... 71

4.4 Model III parameter estimates .................................................................... 72

4.5 Log-likelihood.................................................................................................. 72

4.6 Model I standard errors of parameter estim ates......................................... 72

4.7 Model II standard errors of parameter estimates ...................................... 72

4.8 Model III standard errors of parameter estim ates...................................... 72

4.9 Monte Carlo s tu d y ......................................................................................... 73

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Chapter 1

Introduction

Random field models were introduced in finance by Kennedy (1994, 1997) for the

purpose of modelling forward interest rates and bond prices. This class of models

provides a natural extension of the popular Heath, Jarrow, and Morton (HJM) (1992)

term structure model to infinite factors. A random field is a multiparameter stochastic

process. An n-parameter random field is a stochastic process with random innovations

that vary with respect to each of the n parameters. By analogy with the physical system

described by Brownian motion - a particle moving randomly in space - a two-parameter

random field can be thought of as the random motion of a string, and a three-parameter

random field as the random motion of a membrane.1 Random field term structure

models offer greater flexibility relative to the HJM framework, while at the same time

the number of model parameters can be kept low, by assuming, for example, that the

correlation structure of the random field shocks is a smooth function of the time to

maturity.

Random field models may also be employed to model the Black-Scholes implied

volatility of plain-vanilla European stock options. There exists abundant evidence, which

is discussed in the following chapters, that the volatility of both stock and bond prices

is stochastic, i.e. it cannot be written as a function of the asset price and time only. In

the case of stock options, implied volatility can be used to obtain information about the

unobservable stock price volatility process. In similarity to the HJM model, which fits

the initial forward rate curve, one can build a model that fits the initial implied volatility

surface for a given stock. Using the dynamics of implied volatility, illiquid and exotic

derivatives can be priced, taking the prices of liquid European options as given.

1In this physical analogy the first parameter of the random field represents time, other parameters correspond to the position of a point on the string or the membrane, and the value of the random field may indicate velocity, for example.

In this thesis we develop stochastic volatility models that incorporate random field

innovations for the purpose of pricing stock and interest rate derivatives. The case of

stock options is considered in the next chapter, where we model the implied volatility of

plain-vanilla European stock options as a random field with three parameters: current

time, the maturity date and the exercise price of the corresponding option. The no-

arbitrage condition on asset prices implies that the drift of implied volatility is

endogenously determined. Using this drift restriction, we show that implied volatility

converges to a measure of the instantaneous volatility of the underlying asset price, as

the time to maturity tends to zero, and the exercise price tends to the current stock

price. In this model markets are incomplete due to stochastic volatility in asset prices,

in contrast to the standard Black-Scholes model. We show how the prices of illiquid and

exotic derivatives can be obtained as a function of the stock price and implied volatility.

The random-field implied volatility model allows greater flexibility in fitting observed

volatility smiles than previously studied stochastic volatility models, none of which has

been found to fit observed patterns in implied volatility adequately.2

In chapter 3 we develop a random field model for forward interest rates with

stochastic volatility. We assume that the forward rate volatility function can be de

composed into a deterministic function of the time to maturity and a maturity-independ

ent stochastic process driven by a standard Brownian motion. The latter can be correlated

with the random field shocks driving forward rates. The separability of the forward-rate

volatility function allows us to obtain closed-form solutions for the prices of a number

of standard interest rate derivatives: bond options, interest rate caplets, and interest

rate spread options. In practice, the most common interest rate contingent claims are

LIBOR3 and swap rate derivatives. As a result, LIBOR and swap rate models are more

popular with market practitioners than models for continuously compounded forward

rates. We develop random field models for both forward LIBOR and swap rates with a

separable volatility structure, and derive closed-form solutions for the prices of LIBOR

caplets and swaptions.

Chapter 4 provides an empirical study of random field term structure models. Three

models are considered. In the first model forward rate volatility is a function of the spot

rate and the time to maturity. In the second model forward rate volatility is a function

of the time to maturity and a stochastic process driven by a standard Brownian motion.

In the third model volatility is driven by a random field. In both the second and third

2See for example Das and Sundaram (1999). We discuss the empirical evidence in chapter 2.3London Inter-Bank Offer Rate.

models it is assumed that volatility innovations are uncorrelated with the random field

process driving forward rates. We estimate the models using seven years of daily UK

and US forward rate data, spanning times to maturity between zero and 120 months,

at increments of one month. We use all maturities available in our dataset, in order to

approximate as closely as possible the continuous nature of random fields.

Previous empirical studies of random field term structure models (Longstaff, Santa-

Clara, and Schwartz, 2001a and 2001b) reduce the model to a discrete version with

a small number of factors - only four in the above papers. This makes the model

indistinguishable from finite-factor models, as shown by Kerkhof and Pelsser (2002).

In fact very few empirical studies of high-dimensional models exist, due to the technical

difficulties present in such situations, especially in the case of models with latent variables.

While the number of empirical studies of stochastic volatility models has increased

rapidly in recent years, most of the econometric methods developed so fax are either

inefficient or computationally unfeasible when applied to high-dimensional latent-variable

models.

We estimate the three models by maximizing the likelihood function of the observed

data. In the case of stochastic volatility we perform simulations to integrate volatility

out of the likelihood function, a technique known as Monte Carlo maximum likelihood.

We develop a method of importance sampling that substantially reduces the variance

of the Monte Carlo estimator of the likelihood function, so that the number of required

simulations is low enough to make the estimation process feasible.

9

Chapter 2

A random-field m odel for im plied

volatility

2.1 Introduction

The celebrated Black-Scholes formula (Black and Scholes, 1973) is widely used by

market participants as a benchmark for pricing options. The shortcomings of the model

and its discrepancies with option price data are well documented. The model assumes

that returns on the underlying asset are normally distributed and that the volatility, or

instantaneous variance, of the asset price is constant. However, empirical studies of stock

returns find that unconditional distributions have tails which are thicker than those of the

normal distribution, and that volatility is time-varying and exhibits persistence of shocks.

Some of the earliest evidence is given by Mandelbrot (1963) and Fama (1965).1 Volatility

is also inversely related to the stock price level - the ‘leverage effect’ - (Beckers, 1980),

and exhibits mean reversion in the long run (Merville and Pieptea, 1989).

The Black-Scholes model can be used to solve for the implied volatility of an option

by inverting the pricing formula. In practice, plots of implied volatility against the

exercise price usually form a convex curve (the volatility smile, smirk or skew depending

on whether it is symmetric or not) or, less often, a concave curve (the volatility frown).

Furthermore, implied volatility can be increasing or decreasing in the time to maturity.

These observations provide further evidence against the validity of the Black-Scholes

model’s assumption of constant volatility.2 Numerous models which dispense with

1See also Blattberg and Gonedes (1974), Engle and Mustafa (1992), and Bollerslev, Chou, and Kroner (1992).

2 Apart from nonconstant volatility, other possible explanations for the observed deviations of option price data from the Black-Scholes model are stochastic interest rates, transaction costs, liquidity problems, and feedback effects from dynamic hedging.

10

this assumption have been proposed. Such models can be arranged in two groups:

deterministic volatility (DV) models, where volatility is a function of observable variables,

and stochastic volatility (SV) models, where the stochastic process driving volatility

introduces one or more additional risk factors in the economy.

The earliest DV model (Merton, 1973) extended the Black-Scholes model so that

volatility is a deterministic function of time. Cox and Ross (1976) consider a model

where volatility is proportional to the level of the stock price raised to a power - the

constant elasticity of variance (CEV) diffusion model. Geske (1979), Rubinstein (1983),

and Bensoussan, Crouhy, and Galai (1994) present DV models where option prices are

a function of the value of the firm.

Alternative DV models have been developed in discrete time through the implied

tree approach and GARCH (generalized autoregressive conditional heteroskedasticity)

time series modelling. GARCH option pricing models (Engle and Mustafa, 1992; Duan,

1995; Kallsen and Taqqu, 1998) assume that current volatility is a weighted average of

past squared asset returns and past volatility, with greater weight being assigned to more

recent values. Thus GARCH models can capture volatility clustering - the tendency

of high (or low) volatility values to persist in time. GARCH option pricing models do

not have closed-form solutions however. As a result, simulation methods are usually

employed to compute option prices and hedge ratios.3

Implied tree DV models (Derman and Kani, 1994; Rubinstein, 1994; Dupire, 1994

and 1997) assume that asset price movements form a binomial (or trinomial) tree. Under

this hypothesis the implied risk-neutral distribution of the asset price at maturity is

obtained from the observed prices of European options. Volatility is assumed to be a

deterministic function of the asset price and time, but no assumptions are made about its

functional form, which may be quite complex and unique for every different ending return

distribution. Models of this class fit initial option prices but are not very successful in

predicting future option prices (Dumas, Fleming, and Whaley, 1998).

An important difference between DV and SV models concerns market completeness.

In DV models options are redundant securities while SV models introduce additional risk

factors in the economy. As a result, options are not redundant securities in SV models,

unless one of the following two assumptions is made:

(1) volatility is a traded asset, or

(2) aggregate consumption is uncorrelated with volatility, so that the risk premium asso-

3A related type of DV models are exponentially weighted moments models, proposed by Hobson and Rogers (1998). This class of models assumes that instantaneous volatility is a function of exponentially weighted moments of the historic logarith m s of stock prices.

11

ciated with volatility under the physical measure is zero.

If either (1) or (2) holds, options can be valued using the unique risk-neutral

equivalent martingale measure. Otherwise, markets (excluding options) are incomplete,

and there exist an infinite number of equivalent martingale measures, with each one

corresponding to a particular specification for the volatility risk premium.

A popular type of SV models are diffusion models, which are of the following form

in general:4

dS{t) = pS{t)dt + a ( t)S ( t)d W s (t) (2.1)

da (t ) = a [t, a (t)] dt + b [t, a (t)] dW„ (t) (2.2)

where S (t) denotes the stock price and a (t ) its instantaneous volatility. Ws (t) and

Wa (t) are standard Brownian motions with correlation coefficient p:

cor [dWs (t) , dWa (t)] = p

Table (2.1) shows different specifications for the functions a and b in the volatility

diffusion equation employed in a number of SV models. All of the models in the table

use equation (2.1) as the diffusion equation for stock prices, except Johnson and Shanno

(1987), who modify this equation to:

dS {t) = pS (t) dt + a (t) & (t) dWs (i)

Model a [£, a (£)] b [£, a (<)]

Wiggins (1987) / [a (t)] /3a (t)

Scott (1987), Stein and Stein (1991) 5 [# — a (t)] (3

Johnson and Shanno (1987) rja (t) (3a** (t)

Table 2.1: Volatility diffusion equation parameters

Hull and White (1987), and Heston (1993) model the dynamics of the square of

volatility, v (t) = a 2 (t), as follows:

dv (t) = rjv (t) dt -|- (3v (t) dWv (t) (Hull and White, 1987)

dv (t) = h [9 — v (t)] dt + fiyjv (t)dWv (t) (Heston, 1993)

4Nelson (1990) and Duan (1996, 1997) show that SV diffusion models can be obtained as the continuous-time limit of GARCH models.

12

Wiggins (1987) makes specific assumptions about preferences and model parameters,

such as the asset’s beta coefficient in the capital asset pricing model, and the partial

correlation coefficient between the return on the market and the individual stock’s

volatility. He uses finite differences numerical methods to solve the partial differential

equation for option prices. Johnson and Shanno (1987) assume that there exists an asset

whose price process is perfectly correlated with volatility. They price stock options using

Monte Carlo simulations.

Hull and White (1987), Scott (1987), and Stein and Stein (1991) assume that p = 0,

and that the volatility risk premium is either zero or equal to a constant. Hull and White

(1987) obtain a series approximation for the price of a European call option.5 Scott

(1987) prices options using Monte Carlo simulations. Stein and Stein (1991) derive a

closed-form solution for the stock price distribution. A drawback of the Scott (1987), and

Stein and Stein (1991) models is that the mean-reverting Ornstein-Uhlenbeck process

used to model a (t) can become negative.

Heston (1993) obtains closed-form solutions for European option prices in terms of

the inverse Fourier transform of the characteristic function of the stock price logarithm.

His model allows p to take any value. Thus it can account for the observed negative

correlation between asset returns and volatility. The volatility risk premium is assumed

to be proportional to a2 (t). Schobel and Zhu (1999) use Heston’s method to price options

in the model by Stein and Stein (1991) with non-zero correlation between returns and

volatility. Bakshi, Cao, and Chen (1997), and Scott (1997) also derive option prices using

Heston’s method for SV models which include random jumps in returns, and stochastic

interest rates.

A number of general equilibrium SV models have also been proposed (Bailey and

Stulz, 1989; Amin and Ng, 1993; Bakshi and Chen, 1997; Garcia, Luger, and Renault,

2002). Equilibrium models, however, have the disadvantage that option prices depend

on the model’s assumptions about preferences. Our survey of SV models is by no means

exhaustive. For example, other researchers have considered SV models that can replicate

particular aspects of volatility behaviour, such as random jumps resulting from the

discontinuous arrival of information (see for example Naik, 1993), and long memory

(Comte and Renault, 1998).

Empirical research suggests that SV models explain option prices better than DV

models. Dumas, Fleming, and Whaley (1998) compare the performance of a number of

DV models against an ad hoc (and internally inconsistent, but often used in practice)

5They also present a numerical solution for the case where p ^ 0.

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option pricing procedure where Black-Scholes implied volatilities are smoothed across

exercise prices and maturities, and cure used as an input in the Black-Scholes model. They

find that DV models perform no better in hedging or predicting option prices than this ad

hoc procedure. Their results imply that although the Black-Scholes model is inconsistent

with the data, DV models are unable to improve on option pricing accuracy. Bates (1997)

compares three types of models: CEV (DV) models, SV models with negative correlation

between volatility and returns, and jump-diffusion models with negative-mean jumps in

returns. He finds that CEV models axe less consistent with options market data than

the other two types of models.

Bakshi, Cao, and Chen (2000), and Buraschi and Jackwerth (2001) provide evidence

that options are not redundant securities, particularly after the 1987 crash, and that

additional risk factors need to be added to one-dimensional diffusion models to explain

the data. Lamoureux and Lastrapes (1993) provide evidence against SV models with

zero volatility risk premium. Their data suggest that the volatility risk premium is

time-varying and a decreasing function of the volatility level. Pan (2002) and Benzoni

(2002) also find that the volatility risk premium is statistically significant. A study of the

performance of delta-hedged S&P 500 index option portfolios by Bakshi and Kapadia

(2003) suggests that the market volatility risk premium is negative.

Using S&P 500 index option price data, Nandi (1996) finds that Heston’s (1993)

model performs better than the Black-Scholes model in terms of pricing and hedging. In

a subsequent paper Nandi (1998) finds evidence of nonzero correlation between returns

and volatility using Heston’s (1993) model. Bakshi, Cao, and Chen (1997), and Buraschi

and Jackwerth (2001) find that one-dimensional diffusion models of asset returns are

not improved significantly by the inclusion of stochastic interest rates, relative to the

inclusion of stochastic volatility. Jones (2003) observes that as volatility increases, its

own volatility increases at an even faster rate.6

Recently, market models of implied volatility have been proposed by Schonbucher

(1999) and Ledoit, Santa-Clara, and Yan (2002), in similarity to market models of the

term structure of interest rates, introduced by Brace, Gatarek, and Musiela (1997) and

Jamshidian (1997)7. In this type of models the Black-Scholes implied volatility (IV) is

assumed to follow a continuous-time process with specific dynamics. Ledoit et al. (2002)

model the dynamics of IV, denoted by V (t, s, X), as follows:

dV (t , s, X ) = fiv (t, s, X) dt + oyx (t , s, X) dWi (t) + <jy2 (t, s, X) dW2 (t )

6Garcia, Ghysels and Renault (2004) provide a survey of recent empirical work on stochastic volatility.7Market models of the term structure of interest rates are discussed in section 3.3.

14

where t denotes current time, s the time to maturity of the option, and X the moneyness

of the option, defined as the stock price divided by the exercise price. W\ and W2 axe

independent Brownian motions, with W\ being the Brownian motion driving the stock

price process in equation (2.1). The functions fly (t, s, X), oyx (£, s, X ), and ay2 (t, a, X)

can depend on IV and the stock price in general. Using the discretized dynamics of the

stock price and a Taylor approximation to the standard normal distribution function,

Ledoit et al. (2002) show that as the time to maturity goes to zero, the IV of at-

the-money options converges to the instantaneous volatility of the stock price process.

Using this result, the instantaneous volatility in equation (2.1) can be replaced by

V (t, 0,1). Another important result in this framework is that the IV drift is endogenously

determined through the partial differential equation that a contingent claim must satisfy

and the Black-Scholes partial differential equation, through which IV is defined.

Schonbucher (1999) presents a similar IV model. The dynamics of the stock price

and IV under the risk-neutral measure are assumed to be as follows:

dS (t) = rS (t) dt + cr (£) S (t) dWo (t )N

dV (t,T ,K ) = u (t, T, K )d t + ~f (t, T, K) dW0 (t) + ^ t,„ (t, T, K) dWn (t)n= 1

where T denotes the maturity date and K the exercise price of the option, Wn n = 0,.., N

are Brownian motions, and Wq is orthogonal to Wn for n = 1,.., N. When N = 1 the

model reduces to the Ledoit et al. (2002) model. Brace et al. (2002) show that these IV

models encompass all Markovian stochastic instantaneous volatility models. Brace et al.

(2001) apply this approach to the modelling of the IV of forward LIBOR caplets.8, 9

It would be natural to extend the Schonbucher (1999) and Ledoit et al. (2002)

models so that IV is driven by a random field with three parameters - current time,

the maturity date of the corresponding option, and its exercise price - as suggested by

Ledoit et al. (2002). When IV is driven by a random field, the number of sources of

stochastic shocks in options markets is infinite.10 In this case IV and, hence, option

prices are infinite-dimensional stochastic processes, i.e. they cannot be spanned by a

finite number of independent stochastic processes. It follows that all options are needed

to complete the market in this framework.

8The implied volatility of interest rate caplets is calculated using the Black (1976) model (see chapter3).

9Another type of models that fit initial IV are implied tree models, which were discussed in the context of DV models. Derman and Kani (1998), and Britten-Jones and Neuberger (2000) extend the implied tree approach to include stochastic volatility. Skiadopoulos (2001) provides a survey of this class of models.

10 We exclude degenerate cases where the random field collapses to a finite number of Brownian motions.

15

This result, however, is inconsistent with simple option pricing models, such as those

by Hull and White (1987) and Heston (1993). In these two models, for example, option

prices are spanned by two stochastic processes, and market completeness is achieved

using two securities, such as the stock and an option, or two options (Bajeux-Besnainou

and Rochet, 1996; Romano and Touzi, 1997). We therefore need to examine whether

there exists any specification for the dynamics of the stock price and its instantaneous

volatility that is compatible with IV being modelled as an (infinite-dimensional) random

field. If such a specification does not exist, then the assumption that IV is a random

field necessarily implies that the no-arbitrage condition on option prices does not hold,

possibly due to market imperfections, such as transaction costs and liquidity problems.

The structure of the rest of this chapter is as follows. In section 2.2 the stochastic IV

model is developed and we address the above consistency issues. Under the assumptions

of the model, the entire continuum of plain-vanilla European options for all maturity

dates and exercise prices is required to complete the market. The drift of the IV process

is endogenously determined, as a result of the no-arbitrage condition on option prices

and the Black-Scholes differential equation, through which IV is defined. Using the IV

drift restriction equation we show that IV converges to a measure of the instantaneous

volatility of the stock price, as the time to maturity tends to zero and the exercise

price tends to the current stock price. In section 2.3 we price volatility derivatives

using Heston’s (1993) inverse Fourier transform approach. In section 2.4 we show how

the prices of illiquid and exotic derivatives can be obtained using the dynamics of IV.

Finally, section 2.5 concludes.

2.2 Stochastic implied volatility model

Uncertainty is modelled as a complete filtered probability space (Q, T , F, P), where

P represents the physical measure, and F is assumed to satisfy the usual conditions.n

There exists a non-dividend-paying stock with price process S (t ). Options can be written

on this stock for all exercise prices K and maturity dates T, such that 0 < K < oo and

t < T < oo. There also exists a riskless asset, or bank account, with price process B (t).

We assume that the instantaneous nominal rate of interest r is a positive constant, and

the initial value of the bank account, B (0), is equal to one. Thus, the value of the bank

account at time t is B {t) = ert and its dynamics are:

dB (t) = rB (t ) dt (2.3)

11 Fo contains all the null sets of P, and F is right-continuous.

16

The Black-Scholes formula for the price of a European call option C (t) with exercise

price K and maturity date T at time t, written on the stock S (t), is given by

C(t) = S (i) N (d i ) — e - ^ - ^ K N (d 2) (2.4)

where N (a;) is the standard normal distribution function, and d\, d2 3X6 equal to:

d\ = -n M t .(-r t j ^ _ ) ~ d2 = d x - v V T - t (2.5)v y / T ^ t V '

where v is the instantaneous volatility of the stock price, assumed to be constant, in the

Black-Scholes model.

The implied volatility of an option with price C is defined as the value of v for

which the Black-Scholes formula generates an option price equal to C. It must be

emphasized that the use of Black-Scholes implied volatilities does not imply that the

Black-Scholes model is valid. It is merely a one-to-one relationship between a measure

of volatility and the market price of an option with a given maturity date and exercise

price. For this reason options are usually quoted in terms of their IV instead of their

market price. We model IV because it is a directly observable quantity, in contrast to

the instantaneous volatility of the stock price process. Furthermore, this approach allows

us to fit the initial market prices of plain-vanilla European options. In practice v is a

stochastic process which depends on t , T, and If. Hence it can be represented as a

random field v (t,T ,K). This random field can be visualized as a surface or membrane

evolving stochastically through time.

We define the following function

G tk (S , v,t) = S (t) N (rfi) — e - 'V - ^ K N (d2) (2.6)

which gives the price of a European call option with maturity date T and exercise price

If, as a function of S, v, and t.

The Black-Scholes formula is the solution to the following partial differential equation:

Q G t K , 1 2 o 2< PG t K , 0 9 G t K „ n / odt ^ 2 ~dS^ r ~dS r^ TK = 0 (2-7)

subject to the boundary condition Gtk (T) = max [5 (T) — If, 0].

The partial derivatives of Gt k -, denoted by Greek letters by convention, are as

follows:

A = x j ™ = I r (dl) t = & g t k _ „ ( * )OS ' " a s 2 S v . / T ^ t

V =

17

OGt k

dv Sy/T — tn (di)

d2GTK = S y / T ^ i n { d i ) dld2 &GTk _ d2n(dx)dv2 v dSdv v

_ * 2where n (x ) = 2 is the standard normal density function.

The IV of a put option is equal to the IV of a call option with the same exercise

price and expiration date. This follows from the relationship below, known as the put-call

parity:

C ( t ) - P (t) = S ( t ) — K e -T(T- t)

where P (t) denotes the price of a European put option with the same maturity date and

exercise price as C (t). This relationship holds irrespective of the model that describes

the dynamics of the stock price and volatility, under the assumption that there do not

exist any arbitrage opportunities.

We specify the dynamics of v (t, X, K) as follows:

dkv (t , T, K) = a (t , T, K )d t + ft (t, T, K) dtZ (t, T, K) (2.8)

where Z (t , T, K) is a random field with three parameters, and the subscript t in the

differential operator indicates that the change in the random field is obtained with respect

to changes in parameter t, holding T and K constant.

ASSUMPTION I The random field Z (t, T , K ) satisfies the following conditions:

(i) Z (t,T ,K) is continuous in t, T, K.

(ii) E[dtZ(t,T ,K )] = 0.

(Hi) var[dtZ(t,T,K)\ = dt.

(iv) cor[dtZ (t,T i,K i) ,dtZ (t^T2 ,K 2)] = c(T \,K \,T2,K 2).

(v) dtZ (£i,Ti, K\) and dtZ (t2,T2, K 2) are independent V t\ ^ t 2.

Under the above conditions the correlation structure of the random field Z (£, T, K)

does not depend on t. From (2.6) we obtain that the dynamics of C (t) satisfy the

following equation, using Ito’s lemma:

1 d 2G T K J v , d 2G TK J , OK+ 2 {vt k) + d s d ^ dt {VTK' s) (2-9)

where angles ( ) denote the quadratic variation of a process or the quadratic covariation

of two processes, and vtk (t) = v (t, T, K).

Let <7 (t) denote the instantaneous volatility of the stock price. We assume that o (t)

is stochastic, but we leave it unspecified for the moment. Option prices can be written as

18

a function of the stock price and its instantaneous volatility, as a result of the no-arbitrage

condition that they must satisfy. Defining this function as C (t) = Ht k (•S', a, t), the

dynamics of C (t) axe also given by the following equation, from Ito’s lemma:

dt OS 2 dS2 x ' da 2 da2 w dSda ' ’ '(2.10)

Let $ (t) denote the risk premium associated with a (t), and let p (t) denote the drift

of S (t). The no-arbitrage condition on option prices is given by the following partial

differential equation (Garman, 1976):

E (dC) = ( r C + Od- ^ + S ( » - r ) ^ ) dt (2.11)

where E is the expectation operator.

Since we have two alternative characterizations of option prices, we need to find out

whether the above model is viable, i.e. whether there exist stock price and volatility

processes that can generate option prices and IV which can be described as a random

field. The answer is not obvious, because 5 (£) and a (t) are not functions of T or K,

and, as a result, any stochastic shocks appearing on the right hand side of equation

(2.10) will be functions of t only, while equations (2.8) and (2.9) imply that option prices

change randomly with respect to T and K. A way to make the two equations comparable

arises if the random field can be written as the weighted sum of an infinite number of

independent Brownian motions with weights that vary with respect to T and K.

Definition 2.1 An infinite-dimensional Brownian motion is a sequence W (t ) =

{Wi (t) , i = 0 ,1 ,2 , . . .} of independent standard Brownian motions.

ASSUMPTION II The random field Z (t, T, K) can be written in terms of an infinite

dimensional Brownian motion as follows:oo

(2.12)

where CrKi *s a function o fT , K and i ,12

The dynamics of IV can thus be written as:oo

(2.13)

12 The model can be easily extended to include time-varying Crxi-

19

We assume the following dynamics for the stock price and an associated sequence

of volatilities ai (t):oo oo

dS(t) = H (t) S (t) dt + S ( t ) J 2 Y , K'iai W dW] (*) (2-14)i = l j = 0

dai(t) — 'yi (t)dt + 5i (t)dWi(t) i = 1 ,2 , . . . (2.15)

where (t) , 7j (£) and ^ (£) can be functions of S (t), Oi (t), and t in general, and the nij

are constants.

ASSUMPTION III S (t) , Oi (t), and v ( t ,T ,K ) are strictly positive V t, i, t < T < oo,

0 < K < oo. Furthermore, the parameters in the dynamics of S (t), Oi (t), and

v (t, T , K) satisfy appropriate Lipschitz and linear growth conditions so that there exist

unique and bounded strong solutions to the stochastic differential equations (2.13)-

(2.15).lz

ASSUMPTION IV (i) 3 at least one i such that o ^ 0.

(ii) Si (t) ^ 0 V i , t.

(Hi) fi( t ,T ,K ) ^ 0 V t, t < T < oo, 0 < K < oo.

Let axK (t) = a ( t ,T ,K ) and fiTK (t) = (3(t,T,K). Using equations (2.13)-(2.15)

we expand equations (2.9) and (2.10):

j r , 9Gt k , 9Gtk a , 9Gtk , 1 92Gt k a2 ^dC = ~dt~ + ST1* + dâTK + 2~dS S g ( g ^ )

1 02Gt K r2 sr* >2 . <PGt K qa+ 2 ~ d ^ r PTK g + d S d ^ K SPTK g Qt Ki g ^

1 K 2=0 J = 0 2=1

+ S (2*16)

oo / oo

dt

dC = 9Ht k , 9Htk a , 9Ht k , 1 92Ht k r f V ' f v ' ^“flT + + g T 7i + 2~dS g (g ' )z=l j=0 \z=l /

+ 2 X , + 5 X ,2=1 1 2=1 J = 1

dt

, 0HTk ^ j r r , V"' Z 9HTk , 9 H tk s \ JJxr (rt ^+ - ^ - 5 2 ^ k*oMWo + 2 ^ ( ~Q S~S 2 ^ Kii ai + 53 ) dwi (2*17)

2=1 j = l \ 2=1 3 /

Let A denote the infinite-dimensional matrix with ijth. element CTjKjii wêre

t ^ ^ oo and 0 ^ < oo. When the columns of A are linearly independent, the

13These conditions, which depend on the functional forms of fi (t ) , 7 i(* M < (0 M t , T , K ) , 0 ( t , T , K ) , and CTKH can be found in Da Prato and Zabczyk (1992).

random field Z (t, T, K) cannot be spanned by a finite number of independent stochastic

processes. Let C (t) and v (t) denote two infinite-dimensional vectors consisting of option

prices and implied volatilities, respectively, for all maturity dates t < T < oo and exercise

prices 0 < K < oo, and let <t (t) and W (t) denote the vectors of all cr* (i) and Brownian

motions respectively. The two mappings

Q:(S, v) (S, C)

U : ( S » -> (S, C)

must be consistent with each other. Under assumptions (III) and (IV) the mapping

W —> (S , a) is invertible V t. If the Jacobian of T-L is nonzero V t, then by the inverse

function theorem % is also invertible, and hence all options (and the stock) are needed

to complete the market.

Proposition 2.2 Under assumptions (I)-(IV), if the Jacobian of Ji is nonzero V t, then

the columns of A are linearly independent and the mapping W —> (S,v) is invertible.

PROOF. The Jacobian of H is the determinant of the following matrix:

( 1 9 H t 2K 2 d H T i K i1 dS dS dS

0 N

where 0 is an infinite-dimensional column vector of zeros, and N is the matrix of thefirst-order partial derivatives of H with respect to all (t), i.e. its ijth element is equalfc.

to d£. 3-. Nonsingul;

and (2.17) imply that

OHt- k *to —d£ :J-. Nonsingularity of J is equivalent to nonsingularity of N. Equations (2.16)

9 H t k a _ _ 9 G t k 0 _ , 9 G t k a A~ d S ~ S - -Q S ~ S 2 ^ ,Ki O<ri + -Q ^0TK(TKO

*=1 1=1

9Ht k 0 s r ' , 9Ht k s _ 9Gtk 0 _ , 9Gt k q a i n- g g - S ^ K i j a i + ^ — Sj - - ^ - S + ^ - — pTKC>TKj 1,2,. ..1=1 J i= 1

or in matrix form:

GBA' = Kj0Ci y N'A + S V j (2.18)

where A' denotes the transpose of A; A, B, and G are diagonal matrices with main

diagonals consisting of all Si, and resPec ivelyj y is an infinite-dimensional

column vector with ith element equal to - — and V is an infinite-dimensional

matrix with ijth. element equal to

21

G is nonsingular since is strictly positive for t < T < 0 0 and 0 < K < 00 under

assumption (III). A and B axe also nonsingular by assumption (IV). Using the following

operations on the matrix on the right hand side of equation (2.18):

(a) dividing the first column by ]C £i Kio&i> and

(b) multiplying the first column by Kmj&m and subtracting it from column j + 1

for j = 1,2, . . . ,

we obtain the matrix (Sy N'A). Hence, if N is nonsingular then the columns of A are

linearly independent. Q is always invertible since its Jacobian is equal to:

OGtt /c, Qgr2jr2 1 as as0 G

The following mapping is therefore feasible and invertible:

W -> (S ,a ) (5,C) -> (S ,v ) □

Henceforth we shall assume the following:

ASSUMPTION V The columns of A are linearly independent and the mapping W —>

(S,v) is invertible.

Under the above assumption, TL is also invertible, all options are needed to complete

the market, and we can price any derivative using (S, v). Let K denote the infinite

dimensional matrix with ijth element equal to Kij. If the rank of K is finite, then the

dynamics of the stock price can be written in terms of a finite number of stochastic

processes that are linear combinations of the volatilities (t) and the Brownian motions

Wi (t) and, as a result, the number of options required to complete the market is finite.

Hence, for assumption (V) to hold, the rank of K must be infinite.

ASSUMPTION VI There exists a stochastic discount factor M (t) with the following dy

namics:

= - rd t - £ A,- (t) dW% (f) (2.19)' ’ i= 0

where At- (t) denotes the risk premium associated with Brownian motion (t).

This risk premium can be a function of the stock price and <7* (t), in general. Under

the assumption that there exist no arbitrage opportunities, the product of the discount

factor with the price of a traded asset must be a martingale. The drift of the discount

factor is —r, so that M (t) B (t) is a martingale. By Ito’s lemma, the dynamics of the

22

product of the discount factor with the stock price are:

d(MS) oo / oo

M S = v - r ~ Y l ' £ 2 Kii (TiXj I ( Y l Kii Gi ~ ) dwj

Hence, the drift of the stock price process must satisfy the following condition:oo oo

A* = r + 11, Ki3aiX3 (2-20)j = 0 i — l

In the same way we obtain a restriction on the drift of option prices. Using Ito’s

lemma and equation (2.16), the dynamics of M (t) C (t) are (we suppress the subscripts

T and K):

d (MC)M

d G d G _ 1 <PG _2 v"' I V 9 GaF + dS^+ 2 d&S £ ( EJ = 0 \ t = l /

pfi/i °° °° i pP-n °°^ ^ E E ^ + ^ E ^

j = 0 t = l *=0

( dG dG VE A1 ( dv + dS ^ Kijai)j = 0 \ t = l / .

dt

+E3=0

d G OJk d G ^ \ „d v ^ + a s ^ Kij(7i) ji=i

dWi

For the drift in the above equation to be zero, the following must hold:oo / oo

2 'd v ^ 1 d S ~ ^ ‘"'3~'} dt ' d S ' 2 dS2

_ ^ / dG 0, a G „ ^ \ dG dG 1 a2G „2V^ / v , ,rC + H [ to Kl + gg-5 E ‘ j = -to + dS +2d^S E(E )j —0 V i=l / j-o \»=1 /

+ d v a + 2 d u 2 ^ ^ 0 + d S d w K - i j C j V i j = o j = 0 * = i

The above equation is equivalent to the Garman (1976) equation (2.11), when option

prices are expressed in terms of implied volatilities. Combining this equation with the

Black-Scholes partial differential equation (2.7) we obtain:

oo / oo \ 2

E (E-H ~v '/ dG 0, a G „ A \ dG „, , i a 2G „ ,E + *o5EK = Sc5 0* - r) + 5 rS2 as2 J=0 \i= 1 /

00 00dSdv

j = 0 \ i = l /

dG 1 a2G o2^ >2 c U ,+ a F a + 2 a ^ E < , + 3 5 ^ ^ E E

j = 0 j = 0 i = l

Substituting the Black-Scholes partial derivatives into the above equation, we obtain

the following condition for the IV drift:

ay/T — t = d\( r - t - a 2 E C ? - v r ^ E c , ( f > < * - A,'X J j = 0 j = 0 \ i = l J

t - v j ■3=0

j oo oo Yôo /vÔQ ^ \2 .,2d \ n v- Z ^ 7 = 0 ( z ^ * = l K i j a i ) v

+ T T ^ E E Ki j ff* “3=0i=l 2V y / T ^ t

(2.21)

23

Substituting for d\ from equation (2.5), multiplying throughout by 2vy/T — t and

rearranging, we obtain:14

v = H f)+(r+K)(T“<).1 x{ i H i ) +(r+H (T-t)\-(T-t)}pp>

~ v ['“ ( I ) + ( r + ^ 2)

OO / OO

+ 2 v ( T - t ) P ^ 2 t j ( X l /cy <7*_Ai ) + 2 u a ( T - t ) + 5 ^ Ij=0 \i=l J j=0 \ i=1 y

Taking limits as T —» t and K —> S (t),

o o / o o

Um v2 ->K —*S(t) 3= 0 \» = 1

(2 .22)

Thus we obtain the Ledoit et al. (2002) result for the case of an infinite number

of volatility processes Oi (t).15 The dynamics of v [£, t, S (t)] can be obtained using the

Ito-Venttsel formula (see appendix A):

\ 2

dv [t, £, S (£)] =dv 1 d2v OO I o o

vSjTi? + 5 Kii ai I + s l L l i K (Ptsitsj) Y 1 W *dK 2 d K 2 j=o \i=l j=0 dK i=1

+Q!t5 +dvd T

dvdt + 5 ^ ) dWj (2.23)

i=i

where the partial derivatives are calculated at T = t and K = S (t). In the models of

Schonbucher (1999) and Ledoit et al. (2002) v [t, £, S (£)] is equal to cr(t), and hence the

IV of at-the-money near-the-expiration European options can be used for pricing other

derivatives. In our model v [t, t, S (£)] is not usefiil for pricing derivatives, but it still

provides a measure of the total instantaneous volatility of the stock price process, since

d (S (t)) = S 2 (t) v2 [f, t, S (t)] dt.

2.3 Pricing volatility derivatives

A number of options exchanges have introduced volatility indices which are based

on the implied volatility of options. For example, the Chicago Board Options Exchange

14Note that we assume that S, v , and <7* are strictly positive and bounded stochastic processes V t, i, T, and K .

15 Ledoit et al. (2002) prove that IV converges to the instantaneous volatility of the stock price using discrete-time approximations. Our proof does not rely on approximations, but it requires the assumption that v [£, t, S (£)] is strictly positive.

24

(CBOE) introduced a volatility index, named VIX, in 1993. Originally VIX was con

structed in such a way that its value represented the implied volatility of a hypothetical

at-the-money S&P 100 index option with 30 days to maturity. Its calculation has recently

been changed so that it includes out-of-the-money options and the underlying index is

the S&P 500 index. In March 2004 trading in VIX futures began on the CBOE Futures

Exchange, and there are plans to introduce VIX options in the near future.

Derivatives on volatility allow investors to hedge against and place bets on volatility.

In the absence of volatility derivatives, it is possible, in principle, to trade volatility

through a portfolio which is highly sensitive to volatility, such as a straddle, or through a

dynamic trading strategy. In practice, however, such methods do not replicate volatility

perfectly, while they may also present problems associated with discrete adjustments,

transaction costs, and frequent monitoring needs.

Grunbichler and Longstaff (1996) present a model for valuing futures and options

on volatility. They specify a mean-reverting diffusion process for the volatility V of

returns on a stock index. V can alternatively represent IV. Although V is not a traded

asset, it is assumed that it is possible to construct a non-self-financing portfolio which

exactly replicates V . The volatility risk premium is assumed to be proportional to V.

The price of a European call option on V is obtained in terms of non-central chi-squared

distribution functions and a normal approximation is used to evaluate it. Detemple and

Osakwe (2000) extend this model to cover more specifications for the volatility process,

and value American as well as European options.

In this chapter we assume that there exists only one risky asset in the economy, the

stock (excluding options). The model can be easily extended to include multiple risky

assets driven by Brownian motions which are correlated with each other. We define the

following stochastic process:

= (t)dW,(i) (2.24)' ' 2=0

Given the stochastic discount factor defined by equation (2.19), the change of

measure from the physical to the risk-neutral is obtained from the following equation for

each Brownian motion (t) (Ikeda and Watanabe, 1989, p. 192; Duffie, 1996):

dW® (t) = dWi (<) + ^ — d {Wi (t) ,m(t)) = A, (<) dt + dW, (t) (2.25)

where W® (t) is a standard Brownian motion under the risk-neutral measure Q.

The dynamics of S ( t) , Oi (£), and vtk (t) under the risk-neutral measure are thus

25

given by the following equations:oo oo

— = rdt + E E * , Oi<Wf (2.26)i = 1 j = 0

da, = ('ii ~ \ , S i)dt + 5,dW® i = 1 ,2 ,. . . (2.27)( OO \ OO

— Pt k XZ CrKiî J ^ + Pt k ^ (2.28)i —0 / z=0

and the dynamics of vts = j\Jj2<jLo (Z)Si Kijai)2 are:

dvts =rv n, OO r\ OO o2 OO / OO \ 2

S + * E A (a^)E«^+2& s2E e hj=0 *=1 j=0 \i= l /

+«ts — XZ GtSj^jj = 0

oo / o °° \

^ + E ( A K tsj + a ^ S E dwf (2.29)j'=0 V i=l J

The risk-neutral dynamics of vts can be used to price volatility derivatives such as

volatility swaps and swaptions. An investor who enters a long position in a volatility

swap at time t = 0, with time to maturity T, receives the following payoff at maturity:

•r

* L u ( x ), , d x - K

ro

where u (t) represents the measure of volatility being used for the swap, and the constant

K is the fixed payment. In order that the value of the swap is equal to zero at t = 0,

K must be set equal to the expected value of the floating part under the risk-neutral

measure:

[ u ( x ) d x ) = b [ ^ {u (x)] dx (2-30)where the subscript zero in the expectation operator denotes that the expectation is

taken with respect to the information at time zero. If u = Vts then u represents the

instantaneous volatility of the stock price process. If u — v^s then the security is a

variance swap. Suppose that the dynamics of w = vjs under the risk-neutral measure

are given by the following equation:oo

dujt = 6(p — U3t) dt + y/uJt — v i ’jd W ^ (2.31)j = o

where 6, p, i/, and ipj are constants V j , v > 0, and ujt denotes w (t). Under the above

specification the process has a reflecting barrier at */, so that w* > v > 0 V t. The

expectation in equation (2.30) is equal to:

M = e~6t (o;0 - p) + p

26

Hence, the fixed payment in a variance swap, is:

Ku> = 7fij(1 - e~6T) («D - P) + P

Call and put variance swaptions have the following payoff at maturity T,

respectively:16

max ( f i : ujtdt — Ku, o j call variance swaption

max ( Kw t J 0 ujtdt, 0^ put variance swaption

We define the following variable:

v t = VxdxJo

Hence,

dVt = LJtdt

The price Cq of a call variance swaption at time t = 0 is equal to:

Gg = e~rTF$ [ m a x ^ - t f ^ o ) ]

•£? (Vt Ivt>tkJ - e -rTKuP® (VT > T K W)roo

/ Pq {Vt > x ) dx (2.32)JtKu

where Ivt>tkw is the indicator function and P® (Vt > T K U) denotes the probabihty

that Vt > T K U at time t = 0 under the risk-neutral measure. To derive equation (2.32)

we use the following relationship between the probability density function / (a;) and the

distribution function F (a:) of a random variable a;, obtained by integrating by parts:

e-r rT

.,—rT roo

f x f (x) dx = zF (z) — f F (x ) dx Jo Jo

P0Q (Vt < x) can be obtained in terms of the inverse Fourier transform of the

characteristic function of Vt under the risk-neutral measure, denoted by / (Vt, a;*, t, T; ip)

(Kendall, 1987):17

e -i<pxf (V q^Q jO jT; ip)1 -j rooI § ( V t < x) = - - - I Re2 n J 0 up dcp (2.33)

16Variance swaptions may also be defined in terms of the discrete-time variance of the stock price.17If Vt were a traded asset, we could use a change of measure to convert e~rTE^ (Vt I v t >t k u ) into Vt

multiplied by a probability under the appropriate measure.

27

The price Gq of a put variance swaption at time t = 0 is equal to:

Gq = e~rTI $ [max

l~rT~ f~, - r T rTK,

[k „TP® (Vt < T K W) - E$ (Vt IVt<tkJ ]rTKu/ io (Vr < x ) d x

JoT jo

The characteristic function of Vt is the solution to Kolmogorov’s backward

differential equation:

0 = % + w u,t + ^ f (p~ Wt) + \ ^ i{'Ut~ ' ' ) (2‘34)

where £ = S £ o > subject to the boundary condition

f (Vt ,u,t ,T,T\V>) = e"fVT

The characteristic function solution has the following form:

/ (Vt, Ut, t, T; <p) = gPt+Mtut+ivVt

Substituting this solution into (2.34) we obtain

dD dM , ,/» / x »rO. t \0 = — + —-Ut + t<pu)t + Mte (p - LJt) + M t - (cjt - v) dt dt I

Hence Dt and Mt are the solutions to the following system of ordinary differential

equations, subject to the boundary conditions Dt = 0 and Mt = 0:

dD W/1 _ r0i v dM l l f o,— = - M t9p + Mt — — = -up + Mt0 - - M t £

The solutions are:

Dt = — [iV (T + A) + A (r2 — A2)] In ^1 — Be~A^

- Ar r 5 F & + ( AA2- jVA) A<t + c

where

B = p A eAcr r = a - |V«2 f | — A £

\ 9 . v __ 9pA = t + A A = 2A * = *

and c is a constant determined by the boundary condition for Dt.

28

We can also obtain a closed-form solution for the price of a derivative with the

following payoff at maturity T :

max e(e(L~WT)r- # , ( ) ) (2.35)

where L and K are constants.

The price $o at time t = 0 of the derivative with payoff (2.35) is equal to:

$o = e"rr£ j !![m ax(e(i_u’'r)T- i f , o ) ]

= (e“r ^ / UT< ^ . ln/f) - e - ^ K F * (cur < L - I I n * )

The probabihty in the above equation is obtained from the following inverse Fourier

transform:

where / (ojt, t, T; tp) is the characteristic function of ujt-

Eq (e~TuTIUt<l - 1 . \n k ) is obtained using the following lemma:

Lemma 2.3 Let y denote a random variable and f (tp) its characteristic function,

f (tp) = E (e%{py'). E (eayIy>K), where a and K are constants, is equal to:

f (tp — ia) e~t(pK'

dtp

Itp dtp (2.36)

PROOF. In appendix B.

Thus,

f (c j0,0,T; iT)^ { e - TwTI„T<L- ^ K) =

A In J<)j (U0)O> T;ip + iT)ooRe itp

The characteristic function of dt satisfies Kolmogorov’s backward equation:

dtp

° “ ^ + + ( 237)

subject to the boundary condition

f(ur,T,T-,<p) = ei< ^


/ {ut, t, T; ip) = ec *+w.u>,+ivw,

29

Substituting this solution into equation (2.37) we obtain:

0 = ^ + (tip + Mt) 6 (p -U t ) + (i f + M t f I (w, - u) (2.38)at ctt

subject to the boundary conditions Dt = 0 and Mt = 0.

Equation (2.38) reduces to the following system of ordinary differential equations:

dD -- - (ip + Mt) Op + (i2) ~ 7 (P^ “ i'PC'7)] In ( l - Aeet j + c

where

A = itpe-or 2(9B = tip — — “ Iand c is a constant determined by the boundary condition for Dt.

2.4 Pricing and hedging contingent claims

In incomplete markets there exist an infinite number of equivalent martingale meas

ures (EMMs), with each specification of the volatility risk premium (or premia, in the

case of more than one volatility or factor) corresponding to a particular EMM. When the

volatility risk premium is not zero, contingent claims cannot be perfectly hedged using

only stocks and the riskless asset, in general. Options need to be added to the hedging

portfolio in order to attain a perfect hedge.

In the models of Schonbucher (1999) and Ledoit et al. (2002) all derivatives can

be priced as a function of the stock price and the IV of at-the-money options with zero

time to maturity, which in practice can be approximated using close-to-maturity at-the-

money options. Our model, however, implies that the IV processes of all plain vanilla

European options are needed to price other derivatives, and hedging portfolios for exotic

contingent claims must contain European options of all maturity dates and exercise

prices, in general. Although the construction of such a hedging portfolio is unfeasible in

practice, it may be possible to attain a near-perfect hedge using only a small number of

options which are highly correlated with the contingent claim, since the weight of each

30

option in the hedging portfolio is related to the correlation between the option and the

contingent claim.

This result is very different from the well-known fact that any contingent claim

can be perfectly hedged using any one option and the stock in models with a single

stochastic volatility process, but is similar to the situation in random field models of the

term structure of interest rates, where it may be possible to attain a neax-perfect hedge

for an interest rate contingent claim using only those bonds that are most correlated

with this derivative asset. This feature of random field term structure models is in fact

one of their advantages with respect to the HJM framework, because it is consistent

with market hedging practice, whereas n-factor HJM models, which predict that an

interest rate contingent claim can be hedged using any n bonds, are not. The correlation

between interest rate contingent claims and bonds of different maturities depends on the

maturity characteristics of the derivative asset and, as a result, it is usually easy to select

the appropriate bonds for hedging purposes. In the case of stock options, which have an

exercise price as well as a maturity dimension, however, it is more difficult to ascertain

which options to use in order to hedge an exotic derivative.

In section (2.2) we showed that under the assumptions of our model derivative

securities can be priced as a function of the stock price and all implied volatilities. One

can estimate the dynamics of the IV surface using a small number of parameters by

assuming, for example, that the drift and volatility of IV are smooth functions of the

exercise price and the time to maturity. In general, the price U (t) of a contingent

claim satisfies the following stochastic partial differential equation, which can be solved

numerically, if no closed-form solution exists:

d U d U „ l a ^ U \ 2 dU = ~m+asrS + 2ds*s £ ( J > K«J

aTK — Ptk 2 CrKj^j I dTdK j=o )

j. dÛ °°i k U L L dvTlKldvT2K/ T^ T § C n K ^ n K ^ d K . d T . d K ,

«=0

CpUr r ff TT

+ Jt J k d S d v TK S 0T K § CTKj U Kii CTidrdK

+ X j ( 3 5 5 ^ Kii<Ti + It JK j dW? (2 -39)

31

2.5 Conclusion

In this chapter we develop a model for the Black-Scholes IV of plain-vanilla European

stock options where the stochastic shocks driving IV form a random field with three

parameters: current time, the maturity date and the exercise price of the corresponding

option. The drift of the IV process is endogenously determined under the no-arbitrage

condition on option prices. Using this drift restriction we show that as the time to

maturity tends to zero and the exercise price tends to the current stock price, IV

converges to a measure of the instantaneous volatility of the underlying asset. Under the

assumptions of the model, the entire continuum of plain-vanilla European options for all

maturity dates and exercise prices is required to complete the market.

Using the dynamics of IV, illiquid and exotic derivatives can be priced, taking

the prices of liquid plain-vanilla European options as given. Our model offers greater

flexibility than the diffusion IV models of Schonbucher (1999) and Ledoit et al. (2002)

by having different, but correlated, stochastic shocks for each maturity date and exercise

price. By modelling IV as a random field we can fit the initial IV surface exactly, in

contrast to previous stochastic volatility models. Fitting initial plain vanilla option prices

exactly is an advantage if these options are to be used to hedge more complex derivatives.

There exists much scope for further research in this area, such as the empirical

implementation of the model and comparisons of its performance against other stochastic

volatility models. Furthermore, the model can be extended to include jumps in returns

and/or volatility,18 and stochastic interest rates, and can also be applied to the IV of

foreign currency and interest rate derivatives.

18Eraker, Johannes, and Poison (2003) provide some recent evidence of the existence of jumps in both returns and volatility.

C hapter 3

Random -field m odels o f the term

structure o f interest rates w ith

stochastic volatility

3.1 Introduction

Models of the term structure of interest rates can be classified according to the

variable being modelled: spot rate models, instantaneous forward rate models, and

LIBOR or market models. Spot rate models were the first to be studied (Merton, 1973;

Vasicek, 1977; Cox, Ingersoll, and Ross, 1985). Initial spot rate models were one-factor

models. In such models all forward rates are perfectly correlated. In practice, however,

the correlation coefficient between the returns on bonds of different maturities is less than

one. Having a model that can handle imperfect correlation is especially important when

pricing derivatives which depend on multiple rates with different maturities. Spot rate

models were extended to include multiple factors by adding more sources of stochastic

shocks, such as stochastic volatility and/or imperfectly correlated long-maturity rates.

For example, Brennan and Schwartz (1979) present a two-factor model for the spot rate

and the long-term rate, the latter being defined as the yield on a consol bond which

pays coupons continuously. These two variables are assumed to follow a joint diffusion

process. Rebonato and Cooper (1996) find that the forward-rate correlation structure

generated by two-factor models is also inconsistent with the data, because the correlation

does not decrease fast enough as the difference in maturities rises.

The next generation of term structure models, introduced by Ho and Lee (1986),

were constructed to fit a given initial forward rate curve exactly. Ho and Lee (1986)

33

develop a discrete-time model where the term structure follows a binomial lattice. Heath,

Jarrow, and Morton (HJM) (1992) extend this approach to continuous time. Models of

this type have the advantage of being preference-free, and also offer flexibility with

respect to the number of factors driving the forward rate curve. In a HJM model with n

factors, an interest rate contingent claim can be hedged using any set of n bonds. This,

however, is inconsistent with market practice, where contingent claims tend to be hedged

with bonds of similar maturity.

In recent years there has been increasing interest in random field (or string) models

of the term structure of interest rates. In this class of models, introduced by Kennedy

(1994), each instantaneous forward rate is driven by a different stochastic shock process,

that is correlated with those driving other forward rates. As a result, random field

models are infinite-factor models.1 Models of this type can be easier to calibrate than

HJM models. For example, Santa-Clara and Sornette (2001) calibrate two random

field forward rate models which are completely specified by only three parameters.

Furthermore, random field models have the advantage that they generate hedging portfo

lios for interest rate contingent claims that are more consistent with market practice than

HJM models.2

The simplest random field term structure model assumes that interest rate volatility

is deterministic3 and that interest rate innovations are generated by a Brownian random

field. Kennedy (1994) studies a Gaussian random field model for forward rates, and

derives the restriction on the drift of the instantaneous forward rate process imposed

by the no-arbitrage condition. He also obtains a closed-form solution for the price

of interest rate caplets. In a subsequent paper (Kennedy, 1997) he shows that when

forward rates are generated by a random field and are Gaussian, Markov, and stationary,

the forward rate volatility and correlation functions depend on three parameters only.

Chu (1996) prices a number of interest rate derivatives using the partial differential

equations approach in a random field model for the yield curve, where the random

1 Santa-Clara and Sornette (2001) argue that there is a distinction between infinite-factor HJM models and random field models. Infinite-factor HJM models are in fact random field models when each forward rate is driven by a combination of the factors that is linearly independent of those driving other forward rates - see Filipovic (2000) for example. The class of random field term structure models, however, is wider than the class of infinite-factor HJM models, since the random field innovations need not necessarily be Gaussian.

2In general, bonds of all maturities are needed in order to hedge an interest rate contingent claim in a random field model. If markets are incomplete due to stochastic volatility, options are also needed to construct a perfect hedge. Nevertheless, it may be possible to hedge most of the variability in a contingent claim using one or a few bonds of similar maturity.

3Similarly to chapter 2, forward rate volatility is said to be deterministic if it can be written as a function of interest rates, time, and maturity only. The term stochastic volatility is reserved for the case where volatility introduces additional risk factors in the economy.

34

innovations are generated by a Brownian sheet and the covariance function of bond

returns is a deterministic function of time. Goldstein (2000) extends Kennedy’s (1994)

drift restriction to non-Gaussian random fields and prices bond options using Heston’s

(1993) inverse Fourier transform method. He assumes that forward rate volatility and

correlation are deterministic functions of the spot rate and the time to maturity. Santa-

Clara and Sornette (2001) calibrate forward rate models using two types of random field

shocks: an Ornstein-Uhlenbeck random field and a subexponential correlation random

field. They assume that forward rate volatility is a deterministic function of the time to

maturity.

Numerous empirical studies suggest that the volatility of interest rates is stochastic

- Ball and Torous (1999), Ahn et al. (2003), Dai and Singleton (2003) are some recent

examples. Amin and Morton (1994) test the HJM model with time- and level-dependent

volatility using Eurodollar futures and options data, and find systematic strike-price

and time-to-maturity biases. The implied volatility smiles exhibited by interest rate

derivative prices also offer indirect evidence that interest rate volatility is stochastic.

Jarrow, Li, and Zhao (2003) find that the implied volatility of interest rate caplets4

forms an asymmetric smile when plotted against the cap strike rate. Similarly, Rebonato

(2003) presents implied volatility smiles for the swaptions market. Other recent empirical

studies test for the presence of stochastic volatility by examining whether the prices of

interest rate derivatives are spanned by bond prices. Collin-Dufresne and Goldstein

(2002), Collin-Dufresne, Goldstein, and Jones (2003), and Heidari and Wu (2003) offer

empirical evidence that interest rate derivatives are not redundant securities and that

interest rate volatility risk cannot be hedged by a portfolio consisting of bonds only.5

Spot rate models with stochastic volatility have been widely studied (see for example

Fong and Vasicek (1991), Longstaff and Schwartz (1992), and Chen (1996)). Stochastic

volatility HJM models have received less attention, due to the difficulties they present

in the pricing of interest rate contingent claims and the estimation of model parameters.

Chiarella and Kwon (1999) consider a stochastic volatility HJM model which can be

transformed into a Markovian system. They derive closed-form solutions for discount

bond prices in terms of the state variables in the Markovian system and value bond

options numerically. Andreasen, Collin-Dufresne, and Shi (1998) construct a stochastic

volatility HJM model with the additional assumption that there exists a market in futures

4The implied volatility of interest rate caplets is calculated using the Black (1976) model.5ln contrast to these studies, Fan, Gupta, and Ritchken (2002) find that swaptions can be adequately

hedged using bonds. Collin-Dufresne and Goldstein (2003), however, offer an explanation for this result that is consistent with the presence of unspanned stochastic volatility.

35

on the square of bond price volatility, or that such futures can be perfectly replicated

using traded assets. They show that the latter assumption is satisfied if there exist

futures on yields, which axe available in practice. They price interest rate derivatives

using a non-recombining trinomial tree that fits initial bond and volatility futures prices.

The prices of volatility futures serve to obtain the volatility risk premium.

In the context of a multifactor term structure model, Duffie and Kan (1996) derive

the necessary and sufficient conditions for bond prices to be exponential affine functions

of the factors. One of these conditions is that the spot rate is an affine function

of the factors. Collin-Dufresne and Goldstein (2002) provide some examples of affine

term structure models that generate unspanned stochastic volatility. Duffie, Pan, and

Singleton (2000) derive closed-form solutions for derivative prices in this framework, in

terms of Fourier transforms. Collin-Dufresne and Goldstein (2003) extend this class of

models to include an infinite number of factors. They assume that the continuum of

bond prices and bond-price volatility are the state variables in this setting. Bond prices

axe driven by a Brownian random field and bond price volatility by a standard Brownian

motion.

In this chapter we construct random field models of instantaneous forward, LIBOR,

and swap rates with stochastic volatility, and derive closed-form solutions for the prices

of a number of interest rate derivatives. The rest of this chapter is organized as follows.

Section 3.2 presents a random field model for instantaneous forward rates. Forward rate

volatility is assumed to be the product of a deterministic function of the time to maturity

and a maturity-independent stochastic process driven by a standard Brownian motion.

We allow for non-zero correlation between this Brownian motion and the random field

driving forward rates. The separability of the volatility function enables us to obtain

closed-form solutions for the prices of bond options, interest rate caplets, and interest

rate spread options, using Heston’s (1993) inverse Fourier transform method. In this

model markets are incomplete due to stochastic forward rate volatility. In Section 3.3,

after a brief introduction to market models, we present stochastic volatility random field

models for forward LIBOR and swap rates with separable volatility structure, and price

LIBOR caplets and swaptions. Section 3.4 concludes.

36

3.2 A random-field model for the forward rate with stochastic

volatility

3.2.1 The m odel under the risk-neutral m easure

Uncertainty is modelled as a complete filtered probability space (fi, .F,F, Q), where

F = {Ft\t>o is the Q-augmentation of the natural filtration

Tt = o (Z (it, s ) , W (u) , 0 t)

generated by the random field Z (£, s) and a standard Brownian motion W (t), where t

denotes current time and s the maturity date, t > 0, s >t . We assume that F satisfies

the usual conditions. Q represents the risk-neutral measure.

There exist zero-coupon risk-free bonds for all maturities s >t . The time-t price of

a zero-coupon risk-free bond which pays one currency unit at time s > t is denoted by

P (t, s). The instantaneous forward rate / (t, s) at time t for maturity date s is defined

as the interest rate that can be contracted at time t to be received on an instantaneous

riskless loan at time s:/ ( « , , ) = (3.1)

and

P( t , s ) = e ~ i’ ^ dx (3.2)

The spot rate r (t) is defined as the instantaneous interest rate that is received on

a riskless loan at time t. Thus r(t) = f (t, t). There exists a bank account with price

process B (t) adapted to the filtration {Tt}- We assume that B (0) = 1. The value of

the bank account at time t is:

B (() = eJ'°r‘te (3.3)

Hence,

dB (t) = B (t) r (t) dt (3.4)

The dynamics of the instantaneous forward rate under the risk-neutral measure are

assumed to be as follows:

<kf (t , s) = a (t , s)dt + a (t, s) dtZ (t, s) (3.5)

or, in integral form,

/ (£, s) = f (0,s) + f a (x , s )dx+ [ a (x, s) dxZ (x, s) (3.6)Jo Jo

where the subscript t in the differential operator indicates that the increment is obtained

with respect to time t.

37

ASSUMPTION I The random field Z (t, s) satisfies the following conditions:

(i) Z (t,s) is continuous in t and s.

(ii) E [dtZ (t , 5)] = 0.

(in) var [dtZ (t, 5)] = dt.

(iv) cor[dtZ (M l) ,dtZ (M 2)] = c(si - S2 ) = c(s2 - si).

(v) dfZ(ti ,si) and dtZ fa, S2 ) are independent V t\ ^ t2*

Under the above conditions the correlation structure of the random field Z (t, s) does

not depend on t. The instantaneous volatility of the forward rate, o (t, s), is assumed to

be of the following form:

o(t,s) = y/w(t)g (s - t ) (3.7)

where g (it) is a deterministic function of it and the dynamics of oj (t) are:

du) (t) = [7 — u) (£)] dt + Sy/cj (t)dW (t) (3.8)

where W (t) is a standard Brownian motion, and ft, 7 , and S are constants.

ASSUMPTION II (i) car\dtZ (t , s) , dW (t)] = p (s - t).

(ii) dtZ (ti, s) and dW ( 2) arc independent for t\ ^ t2 -

The separabihty of the volatihty function o (t, s) into a deterministic function of

the time to maturity and a maturity-independent stochastic process implies that the

volatilities of different maturities are perfectly correlated. Although this reduces the

flexibility of the model, it allows us to derive closed-form solutions for the prices of a

number of interest rate derivatives. Greater flexibility could be achieved by specifying a

more general volatility function, such as that proposed by Goldstein (2000):

do2 (t , s) = k (s — t) [0 (s — t) — o2 (t, s)] dt + v (s — t) o (t , s) dtZa (t , s)

where volatility is driven by random field innovations which can be correlated with those

of forward rates:

cor [dtZa (t , s i ) , dtZ (t, s2)] = £ (t, «i, s2)

The above model, however, does not yield closed-form solutions for the prices of interest

rate derivatives and is more difficult to calibrate.

Proposition 3.1 Under assumptions (I), (II), and the no-arbitrage condition on bond

prices, the drift of the forward rate is equal to:

a(t,s) = o(t,s) £ o ( t ,y )c(s - y ) d y (3.9)

38

PROOF. In appendix C.

Under the forward rate volatility specification (3.7), the risk-neutral dynamics of

forward rates and bond prices are:

dtf (£,s) = u (t) g (s - t) £ g (y - 1) c(s - y) dydt + y/u(t)g (s - t) dtZ (t, s)

dt£ ( -y S\ = r( t)d t - y/u{t) f g{y — t) dtZ (£, y) dy5) Jt

Collin-Dufresne and Goldstein (2003) study a random field term structure model

which is a special case of our model with p (s) = 0 V s and Z (t, s) being Gaussian. They

also allow the correlation structure of the random field to be time-varying, and consider

the case where this correlation follows a two-state Markov chain. Kimmel (2004) extends

their model to include multiple latent variables that follow a joint diffusion process.

Under the risk-neutral measure, discounted asset price processes are martingales.

The numeraire used for discounting in this case is the bank account process B (t ). If

instead of the bank account we use the price of a zero-coupon risk-free bond with maturity

T as the numeraire, then there exists a measure, the T -forward measure QT, under which

asset prices discounted by P(t ,T ) are martingales (Geman, El Karoui, and Rochet,

1995).

Proposition 3.2 Under the T-forward measure the random field ZT (t, s) defined by the

equation below satisfies assumption (I):

dtZ T (t, s) = d^Z (t, s) + a( t ,y )c (s — y)dydt (3.10)

PROOF. In appendix D.

Let

= [ 0 (£, y) dtZ (t, y) dy (3.11)m (t) Jt

Then W T (t) defined by the equation below is a Brownian motion under the T-

forward measure:

dWT (t) = dW (t) + ^ — d{W (t) ,m(t))m (t)

= dW (t) + a (t , y)p(y — t) dydt (3.12)

ZT (t,s) and W T (t ) also satisfy assumption (II).

Under the T-forward measure the forward rate with maturity T is a martingale:

dtf (t,T) = a ( t ,T )d tZ T (t ,T)

39

The T-forward measure is useful for pricing interest rate derivatives. Let X (t)

denote the price of an asset at time t. Under the T-forward measure, X (t) discounted

by P (t, T) is a martingale:

X(t)P( t ,T) = E f X{v)

P(v ,T) \for t < v < T

Under the risk-neutral measure, X (t) discounted by B (t) is a martingale:

X(t)B( t)

X (v) for t < v

Prom the previous two equations we obtain:

X(v)B ( t ) ^ l B ( v ) ]

If v — T then

w 1 [B ( T )

3.2.2 Pricing bond options

= P ( t ,T ) E ?

X (T)

X(v)P(v ,T) J

= P { t , T ) E f [X (T)}

for t < v < T (3.13)

(3.14)

Let C (t, m, T, K) denote the time-t price of a European call option with maturity

m > t and exercise price K, written on a bond P {t, T ) with maturity T > m. The

payoff of the option at maturity m is:

C (m, m, T, K) = max [P (m, T) — K , 0] (3.15)

Under the risk-neutral measure, the discounted price process of the call option is a

martingale. Hence,

C (t, m ,T ,K ) = £ ? {e- r r(x>dxIP(m,T)>K [P (m, T) - tf]}

= - B ( t ) KEf?

Using equations (3.13) and (3.14), the above equation becomes:

B (m)

C (t , m, T , K ) = P (t , T) PtQ [P (m, T) > K] - K P (t , m) P^™ [P (m, T) > K]

where P® [P (m, T) > K ] denotes the probability at time t that P (m, T) > K under

measure Qr .

Let 'if) (v, t; (p) denote the characteristic function of In P (m,T) under the v-forward

measure:

40

Using the inverse Fourier transform of the characteristic function, the bond option

price is equal to:

■e-Upi(T , t - ,<p) 'ReC ( t ,m,T,K) = +

-*P(t,ro){§ + Ijf Re Ilf

The dynamics of lnJP ( t,T ) , r (t), and uj (t) under the ^-forward measure axe:

•T r Tdt ]nP (t,T) =

dr (t) =

J g ( u - t ) g ( y - t ) c ( u - y ) d u d y

+w(t) J J g (u — t) g (y — t) c(u — y) dudydt

.___________ rT— y/u (<) J g(u — t) dtZv (t, u) du

> W 9 (°) J g ( y - t ) c ( t - y) dy dt

dt

d f (t, s)

S — t

+y/uj (t)g (0) dZv (t, t)

dco (t) = j/? [7 - w (£)] — 6u) (t) g(y - t ) p(y — t) dy j dt + Sy/u) (t)dWv W

The characteristic function satisfies Kolmogorov’s backward equation. Under the

v-forward measure this equation is:

dtpdt a In P(t ,T)

1+ -

d2tf)

r (t ) ~ 2W (*)A (T>T> *) + u W A (v>T»*)

dtf)2 d [In P (t , T ) f “ W A (T’T ’l) + ™ {/? 117 ~ " (J)1 - C (“ ’ <) >

dtf)dr (t )

d f (t, s)ds

du (t )

-u )( t )g (0) B (v , t)

+ -1

s=t

2 da;2 (t) d2tf)

62u (t) + J1 d2tf)

dr (t) du) (t )

2 dr2 (t)

fig (0) P (0) u) (t)

u ( t )g 2 (0) -

8 \n P ( t ,T )d u ( t )

SPxjj

u (<) 6C(T,t)

din P(t ,T )dr ( t ) u( t)g(Q)B(T,t)

(3.16)

where

/T pw

/ g (u - t) g (y - 1) c(u - y) dudy

/ wg ( y - t ) c ( t - y ) dy

/ wg ( y - t ) p ( y - t ) dy


tf) (u, t; ip) = eD(0+«J(0r(0+M(0^(0+*VlnP(t,r)

(3.17)

(3.18)

(3.19)

41


Substituting this solution into equation (3.16),

dD dJ . . dM . .0 = ^ + T t r { t ) + ^ r uj{ t )+lv

a / (t, s)ds - w (t) g (0) B (v, t)

s = t

+M (t ) [7 - u (£)] - Su (t) C (v, t)} + i M 2 (t) S2u (t) + i J 2 (t) u (t) g2 (0)

—iipj (t) w (t) g (0) B (T, t) + J (t) M (t) Sg (0) p (0) to (t) — iipM (t) uj (t) SC (T, t)

The above equation reduces to the following system of ordinary differential equations:

dMdt = tip -A (T ,T , t ) - A(v,T,t)

+M (t) {P + SC (v, t)} - \ m 2 (t) 52 - \ j 2 (t) g2 (0)

+iipj (t ) g (0) B (T,t) — J (t) M (t) Sg (0) p (0) + iipM (t) SC (T, t)dJdi = ~ lV

S = t

subject to the boundary conditions J (m) = 0, D (m) = 0, M (m) = 0. Given a

particular specification for the functions g (s) , p(s), and c (s), such as the following,

for example,

g(s) = e- p ( s )= p e c(s) = e~K |s| (3.20)

the above differential equations can be solved to obtain a closed-form solution for the

price of a bond option.6

3.2.3 Interest rate caplets

An interest rate caplet with strike rate K for the period [s, s + A] is a Euro

pean call option on the continuously compounded forward rate f A (t, s), which is defined

as:i rs+ A

f A (t,s) = — j f (£, u) du t < s (3.21)

Hence,■i rs+ A

dt fA (t, s) = — j dtf (i, u) du

’The second term in the differential equation for D (t) can be integrated numerically.

42

The option is exercised at time s if f A (s, s) > K, yielding the payoff eAf A(s,s) — eAK

at time s + A. The price of the caplet at time t, Ca (£), is equal to:

Ca (t) = P ( t ,s + A) +A { l f * M >K [eA' a<^> - eAK] }

= P ( t , s + A ) E f +i [eAf ^ I fHS'S)>K]

- P (t , s + A) eAKPp“+* [ /A (s, s) > K]

= P («,, + A) | ** (*»*-*'A> + ^°°Re ■».- iAl j

where the expectation is calculated using lemma 2.3, and ij)A (t, s; ip) denotes the charac

teristic function of f A (s, s) under the forward measure Qs+A. The dynamics of f A (t, s)

and <jJ (t) under Qa+A are:

I rs+ A T r s + Ad t fA (t,s) = — J - aj { t ) g ( u - t ) J g (y — t) c(u — y) dydt

+ a/cj (t)g (u — t) dtZa+A (£, u) | du

du(t) = |^ [ 7 -o j ( t ) \ -8u j ( t ) g ( y - t ) p ( y - t ) d y ^ d t

+ 8 y / ^ d W s+A (t)

The characteristic function of f A (s, s) satisfies Kolmogorov’s backward equation:

+ S ^ t) m [ l - " (<)1 - ** {t) C (* + A ><)} + w g2 (3-22)


ipA (s , s; ip) = etip^ ^ s,s^


1p A ( * , s ; i p ) = e D ( t ) + M ( t ) u ( t ) + v p f ^ ( t ta )

Substituting this solution into equation (3.22) we obtain:

dD dM . u(t) r . . . . . . . . ..0 = — + — w (f)-ty> -2£-[A(s + A ,s + A , s ) - j l ( s + A,ti,s)]

—ip2 ( s + A, s -I- A, s) + iipM ( t ) ( s + A, s) 6 u > ( t )

+M ( t ) {0 [7 - u («)] - Sut ( t ) C(s + A,t)} + M 2 (t) \u> ( t ) S2

43

which reduces to the following system of ordinary differential equations:

^ = - M ( t ) P 7

^ = U (s) + M (i) [W (s) + <SC (s + A, «)] - M2 (t) j

where2

J7(«) = [^4(s + A ,s + A,s) — A(s + A ,u ,s)] + + A ,s + A,s)

W (s) = -* v j ic ( « + A ,s ) i + ^

subject to the boundary conditions D (s) = 0 and M (s) = 0. Given a particular

specification for the functions g (s), p(s), and c (s), the above equations can be solved

to obtain a closed-form solution for the price of a caplet.

3.2.4 Interest rate spread options

A European call interest rate spread option is a security with the following payoff

at maturity T :7

C (fc, Z, T) = max ^ k{T,T)-n{T,T) _ K Qj (3.23)

where r* (Z, T) is defined as

1 f T+ir* (Z, T) = - / / (Z, u) du (3.24)* Jt

In order to price interest rate spread options it is necessary to use a multifactor

model in which interest rates of different maturities can be imperfectly correlated. Let

Rki (Z) = rjfc (Z,T) — ri (Z,T). Under the T-forward measure, the price of a spread option

discounted by the price of the T-maturity discount bond is a martingale:

C(k, l , t ) = P ( ( , T ) l f { [ e J “ fr > - J f ] / j l ,(r)>u }

= P ( t , T ) E ^ { e R^ Th Rkl(TrSDK]

-K P ( t ,T )P ® T [Rkl(T )> lnK ]

7The following derivation can also be applied to the more general case of an option on a basket of yields.

44

where ipR (t, T; (p) is the characteristic function of Rki (T ) imder the T-forward measure.

The dynamics of Rki {t) under QT axe:

u(t) rT+kdRki (t) dudt

dudt

r-i +K r ruk I 9 ( u - t ) \ l g { y - t ) c ( u - y ) d y

f T 9 ~ ^ [JT 9 ^ ~ CÛ ~ dy ^ rT + k

+y/u (t)— I g(u — t) dtZT (t ,u) duk Jt

I rT + l— y/u(t) — I g (u — t) dfZT (t,u) du

* Jt

Kolmogorov’s backward equation for the characteristic function of Rki (T ) is:

n _ <hpR , di>R0 — —— h n „ CJ (t)

dt dRkt (t)

1+ 2 dM JT )u{t)

y A ( T + k , u , T ) - j A ( T + l,u, T) k I

^ A ( T + k ,T + k,T) + ^ A ( T + l ,T + l ,T)

- j - A ( T + k ,T + l,T) kl + dif>R

du (t )

82 cP“tyR+_2 " U duP (t) + dRk, (t) du (t)

{ P h ~ u (<)] - Su (t) C (T, <)}

u (t ) 8 i G (T + k , T ) - \ c ( T + l,T)k I


ipR (t , T ; (p) = eD(t)+M(t)u’(t)+i<PRk‘(t)

subject to the boundary condition:

if>R (T, T; <p) = ei<pRkl{T)

The functions D (t) and M (t) are the solutions to the following system of ordinary

differential equations:

— = -A f( t)0 7dDdt

dMdt = —up

4\ a (T + k, u, T) - \ a (T + 1, u, T)k I

^ A { T + k ,T + k,T) + p A ( T + l ,T + l , T ) - ^ - A ( T + k ,T + l,T)

x2——M 2 (t) — iipM (t) S z \ c ( T + k , T ) - \ c ( T + l,T)k I

subject to the boundary conditions D (T ) = 0 and M (T ) = 0.

45

3.2.5 The m odel under th e physical m easure

The assumption of stochastic forward rate volatility implies that markets are incom

plete.8 As a result, bonds are not enough to span all interest rate contingent claims,

and there exist an infinite number of equivalent martingale measures (EMMs), each

corresponding to a different specification for the volatility risk premium.

We assume that the dynamics of forward rates under the physical measure P are as

follows:

dtf (t , s) =rj (t, s)dt-\-cr (t, s) dtZF (£, s) (3.25)

where ZF (£, s) is a random field satisfying assumption (I) under the physical measure.

The dynamics of bond prices under the physical measure are:

dtP (t , s) P(t ,s) = r(t) d t~ It Tl(t’y)dydt~ f t *by)d t*{ t , y)dy

♦ i / T o (t, y) a (£, x )c(x — y) dxdydt (3.26)

We assume the existence of a stochastic discount factor M (t ) with dynamics:

W i t ) = ~ T ® d t ~ f t ^ ^ y) d^ ^ y) d y ~ (t)dWF (t) (3.27)

where A (£, s) and if) (t) denote the forward rate and volatility risk premia respectively.

Let Pm (t, s) = M (t) P (i, s) denote the time-t deflated price of a pure discount bond

with maturity s. The dynamics of Pm (t , s) are:

~p M( t ’s) = { \ f t f t a ^ y) CT x ) c x ~ y) dxdy~ f V(t ,y )dy^dt

A (£, u) c (u — y) duj a (t, y) dydt

/s pooa (t, y ) p ( y - 1 ) dydt - J A (t, y) dtZF (t , y) dy

-if) (t) dWF (t) - J o (t, y) dtZF (t, y) dy

Under the assumption of no arbitrage opportunities, Pm (t, s) must be a martingale.

Hence,

V (t,y)dy = J (T(t,y)<j( t ,x)c(x-y)dxdy + if)(t)J^ a (t,y) p ( y - t ) dy

+ ft \ft X(t’u)c (u ~ y)du a (t, y) dy

8 Similarly to the previous chapter, markets are incomplete when forward rate volatility is stochastic, unless one of the following conditions holds: (a) forward rate volatility risk can be traded, or(b) aggregate consumption is uncorrelated with forward rate volatility, so that the risk premium associated with volatility under the physical measure is zero.

46

Differentiating with respect to the maturity date s,

77 (t , s) = a ( t,s) J a (t , x) c(x — s) dx + a (t , s) A (t,it) c (u — s) du

+V> (t) (t, s)p(s — t)

The change of measure from the physical to the risk-neutral is therefore given by:

dt + dtZ F(t,s) (3.28)dfZ (t , s) = X (t , u) c (u — s) du + W p(s — t)

We define the process m (t) as follows:

= r \ ( t , y )d tZ (t,») dy + v- (t) (t) (3.29)^ (*/

Under the physical measure, (t) defined below is a standard Brownian motion

and satisfies assumption (II) jointly with ZF (t, s):

<JWr (t) = dW (t) - -J~-d (W (t), m (t))m [t)

dt (3.30)= d W ( t ) - ^ X ( t , y ) p ( y - t ) d y + ip(t)

Hence, the dynamics of uj (t) under the physical measure are:

dh) (t ) = [7 - u (t)] + Sy/u{t) ^ A (f, y) p (y - t) dy + ip (t) ] } dt

+ Sy/u^ jdW r {t) (3.31)

3.3 Random-field models for LIBOR and swap rates with


3.3.1 Introduction

An important drawback of the HJM approach from a practical viewpoint is that

the interest rate being modelled is unobservable. In response to this problem market

models of the term structure emerged. This class of models applies the HJM approach

to observable and discretely compounded interest rates. Initial market models of forward

LIBOR (Brace, Gatarek, and Musiela, 1997; Musiela and Rutkowski, 1997; Miltersen,

Sandmann, and Sondermann, 1997) and swap rates (Jamshidian, 1997) assume that

LIBOR and swap rates, respectively, follow a lognormal process under the corresponding

forward measure. These LIBOR and swap rate models are in fact mutually inconsistent,

because these two types of interest rates cannot be simultaneously lognormally distributed

under the no-arbitrage condition.

47

Under the lognormality assumption, the prices for caplets and swaptions are given

to be lognormaUy distributed. The use of the Black formula by traders for pricing

interest rate derivatives was at first considered erroneous, as forward rates for different

arbitrage opportunities arising. However, it was subsequently shown by Brace, Gatarek,

is assumed to be lognormal under the corresponding forward measure. Similarly to the

of caplets and swaptions.

The existence of implied volatility smiles and skews in interest rate derivatives

markets has prompted the appearance of market models with stochastic volatility.

Andersen and Brotherton-Ratcliffe (2001), Joshi and Rebonato (2001), and Andersen

and Andreasen (2002) extend the standard LIBOR market model to include stochastic

volatility. Jarrow, Li, and Zhao (2003) compare empirically the standard lognormal

LIBOR market model against the stochastic volatility model of Andersen and Brotherton-

Ratcliffe (2001), the market model with jumps of Glasserman and Kou (2003),9 and a

combination of the two. They find that while the models with stochastic volatility

and/or jumps perform better than the standard LIBOR market model, none of the

models considered can adequately explain the entire cap implied volatility smile. In this

section we construct random field market models for forward LIBOR and swap rates

with stochastic volatility.

3.3.2 LIBO R m odel

The forward LIBOR rate L (t ,T j) is defined as the interest rate that is earned on a

risk-free loan between dates Tj and Tj+1:

by a formula which is very similar to the Black-Scholes formula for stock options and is

known as the Black formula. This formula was originally derived by Black (1976) as the

price of an option on commodity futures when the change in the futures price is assumed

maturities cannot all be simultaneously lognormal under the same measure without

and Musiela (1997) that market practice is consistent with theory if each forward rate

Black-Scholes formula, the Black formula can be inverted to obtain the implied volatility

1 + a,Tj L (t , Tj) — e',Stj+1 ffa)dx t < T j < Tj+1 (3.32)

where ay. is the day count fraction for the time period [Tj,Tj+i\. Hence,

(3.33)

9In their model volatility is deterministic but there are jumps in interest rates.

48

Let U (t) = $ +1 f (t,x) dx. Then L (t,Tj) = [euW - l] andl 3

- T i -

dU (t) = I dtf (t , x) dx

= l + aTjL(t, Tj)dtL(t,Tj) = - ' ^ ■ v' 7 -J~ dU(t) + - d ( U ( t ) )

O-TjT,

1

2

dU (t) + \ d ( U (t)) = [ 3+1 a (t, x ) dtZ (t, x) dx2 J T j

f T j + i r x

+ I <7 (t,x) I o (t , y)c(x — y) dydxdt J T j Jt1 f Ti+1 [ Ti+1

+ - / / a (t, x) (T (t, y) c ( x - y ) dydxdt1 J T j J T j

Let

/*Ti+ 1 ™(f, Tj) = L (t, Tj) 7j (f) / <7" (t, x) d<ZT>+1 (t, x) dx (3.35)

J T i

l + aTjL(t,Tj)W = „ r / i T (3-34)aTjL(t,Tj)

The dynamics of L (t, Tj) under the forward measure QTj+1 are:

"Tj+1

'Tj

Hence L (t, Tj) is a martingale under QTj+1. The standard LIBOR market model

assumes that forward LIBOR rates have the following dynamics under the corresponding

forward measure:

dtL (t , Tj) = a (t , Tj) L (t, Tj) dWTj+1 (t )

where cr (t,Tj) is a deterministic function of t and Tj, and W Tj+1 (t) is a standard

Brownian motion under Q ^ 1. Andersen and Brotherton-Ratcliffe (2001), and Andersen

and Andreasen (2002) extend the standard LIBOR market model to include stochastic

volatility. They assume that a (£, Tj) has a stochastic component V (t) which is modelled

as follows:

dV{t) = K [ 0 - V (*)]dt + rjf [V (t)]dWyi+1 (t)

where Wyj+1 (t) is a standard Brownian motion under Q^+ 1 which is independent of

the Brownian motion driving LIBOR rates. Andersen and Brotherton-Ratcliffe (2001)

derive the price of a caplet using the Taylor series expansion technique of Hull and White

(1987). Andersen and Andreasen (2002) price caplets and swaptions using Heston’s

(1993) inverse Fourier transform method.

In this section we assume that the dynamics of L (t, Tj) under the forward measure

Q7 1 are:10

dtL (t, Tj) = L (t, Tj) \ fw jj)g (Tj - 1) dtZ T>^ (t, Tj) (3.36)

JThe random field Z (t, s) here is not related to the one used earlier.

49

where g (u) is a deterministic function of it, and w (t) is a stochastic process with the

following dynamics under QU+i :

dw(t) = [7 - w (t)] dt + Sy/w (t)dWTi+x (t) (3.37)

where W T*+1 (t) is a standard Brownian motion under QTj+1, and Z Ti+1 (t,Tj), W Tj+1 (t )

satisfy assumptions (I) and (II) under QTj+1 . This specification of the dynamics of

LIBOR rates is not compatible with the forward rate model described earlier, but it

allows us to obtain a closed-form solution for the price of a LIBOR caplet.

A LIBOR caplet with strike rate K for the period [Tj,Tj+1] is a European call option

on the LIBOR rate L (t ,T j). The payoff of the caplet at time TJ+i is equal to:

Cd (Tj+i) = aTj max [L (‘Tj, Tj) - K, 0]

Under the forward measure (Q)U+i the price of the caplet at time t is:

Ca (t) = P ( t ,TJ+l)aT]E f ’+' - K ] }

= P( t ,TJ+l)aTjl f ’+' (Tj,Tj) Ii^Tj,7})>jr]

- P ( t ,T J+1)aTjK p f ’+1 [L(Tj,Tj) > K]

, P ( ,,r w ) a r i | * ^ 3 i z f l +

where ipL (t,Tj‘,ip) denotes the characteristic function of InL(Tj,Tj) under measure

QTj'+i. The dynamics of lnL (t , Tj) under QU'+i are:

dt In L (t , Tj) = y/w(t)g (Tj — t) dtZTj+1 (t, Tj) — (t ) g2 (Tj — t) dt (3.38)

Kolmogorov’s backward equation for the characteristic function is:

di})Ld i b L x d ( t ) 9 .

0 = i r + ^ 3 ^ ~ t) d [ \a L ( t ,T j ) f S in L(t,T j)

+ (t) + (<) 3 (T> - t)p{T> - *> (339)


(t,Tj\cp) = c°(0+^W®(t)-Hv in £(*,:!})


ipL (Tj, Tj-, (p) = e i<plnL(TiTj)

50

Substituting this solution into equation (3.39), we obtain the following system of

ordinary differential equations, subject to the boundary conditions D (Tj) = 0 and

which yield a closed-form solution for the price of a caplet, given a particular specification

for the functions g (u) and p (u).

3.3.3 Swap rate m odel

An interest rate swap is a contract where two parties agree to exchange interest

rate payments. One party pays fixed interest while the other pays floating interest. A

floating interest payment made at time Tj+i is based on the LIBOR rate fixed at time

T j, and is equal to a ^ L (Tt-, T j) . The par swap rate F is defined as the fixed interest rate

for which the present value of the swap is equal to zero. Let T n denote the date at which

the first LIBOR rate is fixed, and Tn+1, . . . , T/v the dates at which swap payments are

exchanged. The forward par swap rate is given by:

q h + i ,n js forward swap measure (Jamshidian, 1997). The par swap rate F (t, Tn, Tn)

is also a martingale under Qn+1»JV.

option expires at time Tn, and if it is exercised, the swap starts at time T n and ends at

Tn - The present value of the swap is:

~ = ^ g 2 [Tj — t)(i + ip) + M(t)fS — ^ -M 2 (t) — itpM (t) Sg (Tj — t)p(Tj — t)

(3.40)

i= n -f l

Let Qn+1,j/v denote the measure under which asset prices discounted by the following

numeraire are martingales:

N

i= n + 1

A swaption gives the right to enter into a swap with fixed rate K at expiration. Let

S ( i ,T n ,T /v ) denote the time-i price of a payer swaption, i.e. a European option on a

swap in which the buyer of the option pays fixed interest if the option is exercised. The

NV (t,T n,TN) = P (t ,T n) - P ( t , T N) - K Y ,

* = n + l

51

Thus a swaption is an option on a portfolio of bonds. The price of the swaption at

time t is:

S ( t ,T n ,T iv ) F(Tn,Tn,TN)> K

D (t,Tn+i,7V) Bj1 + {[F(Tn,T„,Tjv) - if] lF(Tn,T„,TN)>t:}

D (t,Tn+uTN) E ^ +1'N [F(Tn,Tn,T„) lF (T n,Tn,TN)> K \

- K D ( t ,T n+l,TN) l f ' +1N [F(Tn,Tn,TN) > K]

where ipF (£,Tn,T/v; <p) denotes the characteristic function of InF (Tn,Tn,T/v) under

dtF (t,T„, Tn ) = F (t , T„,Tn ) ^ { t ) g (T„ - t , T N - t) dtZ n+l'N (t,Tn,TN) (3.42)

where W n+1,N (t) is a standard Brownian motion under Qn+1,7V, and Zn+1,N (£, Tn,T/y),

iyn+i,N ^ satisfy assumptions (I) and (II) under Qn+1’N.

Kolmogorov’s backward equation for ipF (t, Tn, T^; ip) and its solution axe similar to

those derived in the previous section for LIBOR caplets:

l/)F (t,Tn,TN;ip) = e D(t)+M(t)v(t)+i<p]nF(t,Tn ,TN)

where D (t) and M (t ) are the solutions to the following system of ordinary differential

equations:

qti+i,jv w e assume that the dynamics of the forward par swap rate under Qn+1>N

are as follows:

where g (r, y) is a deterministic function of x and y, and v (t) is a stochastic process with

dynamics

dv (t) = /3 [7 — v (£)] dt + 5y/v (t)dWn+1,N (t ) (3.43)

dDdt

dM

- M (t) £7

c 2| g2 (T„ - 1, Tn - 1) (i + <p) + M (i) ( S - - M 2 (t )

—iipM (t) Sg (Tn - t , T N - t ) p (Tn - t , T N - 1)dt

subject to the boundary conditions D (Tn) = 0 and M (Tn) = 0.

52

3.4 Conclusion

In this chapter we construct random field models of the term structure of interest

rates with stochastic volatility. We assume that the volatility of interest rates can

be decomposed into a deterministic function of the time to maturity and a maturity-

independent stochastic process driven by a standard Brownian motion. We allow for

non-zero correlation between this Brownian motion and the random field driving interest

rates. The separability of the volatility function enables us to obtain closed-form solutions

for the prices of bond options, interest rate caplets, interest rate spread options, and

swaptions using Heston’s (1993) inverse Fourier transform method.

We present three different models for instantaneous forward rates, forward LIBOR,

and swap rates. These models fit initial instantaneous forward rate, forward LIBOR,

and swap rate curves, respectively. While these models are mutually inconsistent, each

one can be useful for valuing different types of interest rate contingent claims. Having a

model with simple dynamics for the interest rate that is relevant for pricing a particular

derivative has advantages in terms of ease of calibration and accuracy in pricing. These

factors can be more important than having a single model for all securities, in practice, as

inconsistencies between prices in different markets may exist due to market imperfections.

53

C hapter 4

Estim ation o f random-field

m odels o f the forward rate w ith


4.1 Introduction

Empirical research on the term structure of interest rates has largely focused on

models with a small number of factors, due to the difficulties encountered in estimating

large-scale multifactor models. A popular multifactor term structure model is the Heath,

Jarrow, and Morton (1992) (HJM) model. Pearson and Zhou (1999) estimate non-

parametrically the forward rate volatility functions in a HJM term structure model with

one and two factors. They find that the relationship between forward rate volatility and

the level of forward rates is weak, while there is a strong relationship between volatility

and the slope of the forward rate curve for low and moderate forward rate levels. Jeffrey

et al. (2003) develop a non-parametric estimator for the volatility function of the yield

curve in a HJM model which takes into account yield curve measurement errors, and

they implement this estimator using HJM models with one and two factors. Chiarella,

Pasquali, and Runggaldier (2001) propose a Bayesian filtering algorithm for estimating

a HJM model where forward rate volatility is a function of the time to maturity, the

spot rate, and one fixed maturity forward rate. In all of these three empirical papers the

volatility of forward rates is assumed to be deterministic.

Random field forward rate models extend the HJM framework to an infinite number

of factors. As this is a relatively recent approach of modelling the term structure of

interest rates, there exist few empirical studies of such models. Santa-Clara and Sornette

54

(2001) calibrate a deterministic volatility random field forward rate model using two

types of random field shocks: an Ornstein-Uhlenbeck random field and a subexponential

correlation random field. In each case the model is described by three parameters,

which are estimated by minimizing the sum of squared differences between empirical

and model-implied bond price covariances for all available maturities.

Longstaff, Santa-Clara, and Schwartz (2001a) use a random field LIBOR model

with deterministic volatility to examine the relative valuation of caps and swaptions.

They find that long-dated swaptions are undervalued relative to other swaptions and

cap prices differ from those implied by swaption prices under the no-arbitrage condition.

They suggest that a model which allows for a time-varying covariance structure may

explain better the observed relative pricing of caps and swaptions. In another paper

Longstaff et al (2001b) calculate the loss in using a single-factor model to price American

swaptions when the true model is a random field model, and they find that using a

misspecified model can be very costly in this case. In both of these studies Longstaff

et al use principal components analysis to decompose the forward LIBOR covariance

matrix into a number of factors. The number of significant factors found is four and,

as a result, only four factors are used to approximate the original infinite-factor random

field LIBOR model by a finite-factor one. This approximation, however, reduces their

random field LIBOR model to the standard LIBOR market model introduced by Brace

et al (1997). In fact, Kerkhof and Pelsser (2002) show that when random field LIBOR

models are reduced to a finite number of factors, they become observationally equivalent

to the standard LIBOR market model. By observational equivalence Kerkhof and Pelsser

(2002) mean that

“for every specification in the class of discrete string models one can find a specification in the class of market models with the same probabilistic properties and vice versa.”

In this chapter we estimate random field models of the forward rate while maintaining

their infinite-factor structure. Three random field term structure models are considered.

In model I forward rate volatility is a function of the spot rate and the time to maturity.

In model II forward rate volatility is the product of a deterministic function of the time

to maturity and a stochastic process driven by a Brownian motion. In model III volatility

is driven by a random field. In the two stochastic volatility models we assume that the

stochastic process driving volatility is uncorrelated with the random field driving forward

rates. The number of model parameters is five, seven, and ten in models I, II, and III

respectively. Each model is estimated using seven years of daily UK and US forward

55

rate data, spanning times to maturity between zero and 120 months at increments of

one month. Model I can be estimated using standard estimation techniques, such as the

generalized method of moments and maximum likelihood. The combination of a large

number of factors with stochastic volatility in models II and III, however, requires a

more careful selection of the estimation method.

A number of different approaches have been developed for estimating stochastic

volatility models: applications of the generalized method of moments (GMM) introduced

by Hansen (1982) (Wiggins, 1987; Scott, 1987; Melino and Turnbull, 1990; Andersen,

1994; Ho, Perraudin, and Sorensen, 1996), Monte Carlo maximum likelihood (MCML)

(Danielsson and Richard, 1993), nonparametric simulated maximum likelihood

(Fermanian and Salanie, 2004), the simulation-based method of moments (SMM) (Duffie

and Singleton, 1993; Andersen and Sorensen, 1996), the indirect inference approach

(Gourieroux, Monfort, and Renault, 1993), the efficient method of moments (EMM)

(Gallant and Tauchen, 1996; Andersen and Lund, 1997; Gallant and Long, 1997),

Kalman filter techniques (Harvey, Ruiz, and Shephard, 1994), and Markov chain Monte

Carlo methods (MCMC) (Jacquier, Poison, and Rossi, 1994).

Applications of the above methods have concentrated on models with a low number

of latent variables, due to the difficulties presented by high-dimensional stochastic volat

ility models. Nonparametric simulated maximum likelihood is unfeasible for multivariate

models of very high dimension - 121 in our case - as the sample size required for each

simulation becomes prohibitively large.1 Method-of-moments-based techniques, such

as GMM and SMM, also become computationally unfeasible as the number of latent

variables and model parameters increases, and axe inefficient, in general, unless they

involve a continuum of moments (Carrasco and Florens, 2002) or other special cases.

Recent advances in GMM estimation based on the characteristic function (Singleton,

2001; Jiang and Knight, 2002; Chacko and Viceira, 2003) require the characteristic

function of the observations to be known in closed form, which in our case is not

available in any of the models we consider. Carrasco et al. (2004) extend this approach

to cases where the characteristic function is not known analytically, by estimating it

using simulations. In this way, however, discretization bias is introduced, while the

computational cost increases exponentially with the dimension of the model.

MCMC estimation methods offer an alternative avenue for h andling complex

stochastic volatility models. They also allow the estimation of stochastic volatility and

the comparison of non-nested models using the Bayes factor. The success of MCMC

1See Silverman (1986), pp. 93-94, and Scott (1992), chapter 7.

56

methods, however, depends on the rate of convergence of the estimated parameters to

the true values of the model parameters. As the number of latent variables increases -

which in our case is equal to the number of observed variables, 121, in model III - the

number of iterations required for convergence to occur can become unfeasibly large.2

In recent years a number of studies have addressed the problem of discretization

bias in the estimation of continuous-time models using discrete-time data. Ait-Sahalia

(2002) estimates fully-observed continuous-time diffusions through maximum likelihood,

by approximating the true likelihood function using Hermite polynomials. Ait-Sahalia

and Kimmel (2004) apply this method to the estimation of stochastic volatility stock

price models, by assuming that stock price volatility can be approximated by the implied

volatility of close-to-maturity at-the-money options, and using closed-form expansions for

the joint likelihood function of the stock price and its volatility. Pedersen (1995), Brandt

and Sant a-Clara (2002), and Durham and Gallant (2002) develop a method of maximum

likelihood estimation of diffusions where the true likelihood function is approximated

through simulations that serve to integrate out unobserved states between observations.

This technique, known as simulated maximum likelihood, can also be extended to latent

variable models. A related approach is suggested by Elerian, Chib, and Shephard

(2001), and Eraker (2001), who employ MCMC methods to estimate diffusions, using

simulations to generate additional data between observations and integrating them out of

the likelihood function. Simulation-based methods, however, increase the computation

time of the estimation process, which can make the procedure impractical for high

dimensional models. Using high-frequency data may be a better way to reduce

discretization bias in such cases. We use daily data in order to minimize the discretization

bias.

We estimate model I using maximum likelihood,3 and models II and III using

MCML, adopting a standard Euler discretization scheme. In MCML volatility is integ

rated out of the likelihood function using simulations. The accuracy of the likelihood

function estimate depends on the variance of the Monte Carlo estimator. There exist a

number of variance reduction techniques, one of which is importance sampling. In this

method volatility samples are not drawn from the volatility density implied by the model,

but from the importance sampling density, which is defined in such a way that the ratio

2Chib et al. (2002) estimate a high-dimensional stochastic volatility model using MCMC methods, but their technique relies on the particular specification of the model, which allows them to speed up the sampling process through a transformation of the model into a number of low-dimensional independent models. In our case, however, such a simplifying transformation is not possible.

3The asymptotic efficiency of maximum likelihood estimation is well-known. See, for example, Rao (1973).

57

of the joint density of forward rates and volatility to the importance sampling density

is almost constant for all volatility values. A careful construction of the importance

sampling density can result in a substantial reduction in the variance of the Monte Carlo

estimator. The rest of this chapter is organized as follows: Section 4.2 describes the

random field forward rate models to be estimated. Section 4.3 explains the estimation

procedure for each model. Section 4.4 describes the dataset and section 4.5 presents

the empirical results. Finally, section 4.6 concludes and provides suggestions for further

research.

4.2 The models

Uncertainty is modelled as a complete filtered probability space F, Q), where

Q represents the risk-neutral measure and F = {Tt}t>0 is the Q-augmentation of the

following natural filtration in each model:

model I

model II

model III

Tt = <J (Zf (it, s) , 0 0)

Tt = o (Zf (u, s) , W (u) , 0 0)

Tt — o (Zf (u, s) , Za (u, s) , 0 0)

where Z f (t, s) and Z a (t,s) are random fields, W (t ) is a standard Brownian motion, t

denotes current time and s the time to maturity, t, s > 0 . We assume that F satisfies

the usual conditions.

At each time t there exist zero-coupon risk-free bonds for all times to maturity

s > 0. The time-t price of a zero-coupon risk-free bond which pays one currency unit at

time t + s is denoted by P (t,t + s). The instantaneous forward rate / at time t for time

to maturity s is defined as:

/(«,») = - ahtJ>% * + a) (4.1)

and the spot rate at time t is r ( t ) = / (t, 0). We assume that the dynamics of the

instantaneous forward rate under the risk-neutral measure are as follows:

dtf (£, s) = a (tj s) d tp a (t , s) dtZf (t , s) (4.2)

Table 4.1 presents the specification for the volatility function cr (t, s) in each model.

In models I and II g (s) denotes a deterministic function of the time to maturity s.

Model I is a random field forward rate model with deterministic volatility that belongs

to the class considered by Goldstein (2000). In model II forward rate volatility is the

58

Model Volatility function

I <r(t,s) = g(s)y /r{ t)

II 0 (t, s) = a (t) g (s)

diner (t) = C [0 — In<7 (£)] dt + r}dW (t)

III dt lner (t, s) = C («) [0 (s) — In a (t , s)] dt + r} (s) dtZa (t, s)

Table 4.1: Volatility function specification

product of a deterministic function of the time to maturity and a maturity-independent

stochastic process o (t) driven by a standard Brownian motion. In model III volatility

is driven by a random field Za (t, s), assumed to be Gaussian. In all three models the

random field Zf (t, s) driving forward rates is also assumed to be Gaussian. In models

II and III we assume that the random shocks in volatility are uncorrelated with those of

forward rates:

ASSUMPTION I (i) dtZf (t , s ) and dW (t ) are independent

(ii) d tZf(t ,s i) and dtZa (t, S2 ) are independent V si, 52-

ASSUMPTION II The random fields Z{ (£, s), i = / , cr, satisfy the following conditions:

(i) Zi(t,s) is continuous in t and s.

(ii) E[dtZi (t,s)] = 0.

(in) var [dtZi (t, s)] = dt.

(iv) cor [dfZi (t , s i ) , dtZi (f, s 2)] = Q (« i , s 2).

(v) dtZi(t i ,si) and dtZi (t2, S2) are independent V ti ^ t2.

There exists a bank account with price process B (t ) adapted to the filtration {JFf},

and we take B (0) to be equal to one. The value of the bank account at time t is

B (t) = efor(x)dx. Hence, dB (t ) = B ( t ) r (t) dt.

Under the no-arbitrage condition on bond prices, the forward rate drift is equal to

(see appendix E):

a (t, s) = s -|-o (t ,s ) [ o (t , x) c (s, x) dx (4.3)us Jq

where c (s, y) = C/(s ,y) .

59

Hence, the no-arbitrage dynamics of forward rates under the risk-neutral measure

are:

d f( t , s)dtf (t, s) = + <j(t,s) f a ( t )y)c(s,y)dy

Jodt + a (t, s) dtZf (t, s) (4.4)

ds

Denoting the physical measure by P and the forward rate drift under this measure

by n (t, s), the forward rate dynamics under P are:

dtf (i, s) = fj, (£, s)dt + o (t , s) dtZF (t , s) (4.5)

where Z j (t, s) is a random field satisfying assumption (II) under P. Using the results in

appendix E we obtain the dynamics of bond prices under the physical measure:

dtP (t, u) r 1 ru~l ru~l^ ru —t ru —t

p{ tu) ~ \f { t 'u - t) + 2 j 0 J0 a (^x)a (^y)c(x y)dxdyr u — t r u — t

j f i (£, x) dxdt — J cr(t, x) dtZJ* (£, x) dx

dt

To derive the functional form of the drift fi (t, s) implied by the no-arbitrage condi

tion, we assume the existence of a stochastic discount factor M (t ) with the following

dynamics in each model:

model I : ^ ^ = ~ r W dt ~ J * (*> y) dtZF (t , y) dy

model II : ^ ^ = - r (t ) dt - J A (t, y) dtZJ (t , y)dy~4> (t ) dWF (t)

dM (t) r°° r°°model III : — = - r (t) dt - A (t , y) dtZF (t , y) dy - if) (t , y) dtZF (t, y) dy

where A (£, s) denotes the forward rate risk premium, and if) (£), (t, s ) the volatility risk

premia in models II and III respectively. For no arbitrage opportunities to exist, the

product M (t ) P (t, s) must be a martingale. In model III the dynamics of Pm (t, s) =

M (t ) P (t , s) are:

dtPM (t, s) ^* — 4“ = ~ r ( ) d t — f n (t, x) dxdt + /* f a (£, x) A (t , y) c (x, y) dxdydtP m (t , s) Jo J x=o Jy=o

r u — t r u — t

f ( t , u - t ) + - / a {t,x )a (t ,y )c (x ,y )dxdyr o o r o o

~ y) <kZF (t , y)dy — j A (t, y) dtZ j (t, y) dyJo Jo

r u — t

— I a (t, x) d*Zj (£, x) dxJo

Hence,r u — t j r u — t r u — t

/ /x(t,x)dx = / ( t , u - t ) + - / / a ( t,x )a (t ,y )c (x ,y )d xd yJo * Jo Jo

r u — t r o o

- r { t ) + / o- (£, x) A (t, y) c (x, y) dxdyJx= 0 J y = 0

r ra roog (s) I r (t) j g(x)c (x , s) dx + y/r (t ) J A (t , it) c (it, s) du

y (a;) c (a;, s) dx + o (t ) J A (£, it) c (it, s) ditj

60

Let s = u — t. Differentiating the above equation with respect to u,

d f (t s) f s f°°/i (t , s) = —^ ----\ - < j ( t , s ) G (£, x) c (s , a;) da; + cr (t, s) / A (t , y) c (s, y) dy0s 7o Jo

Hence, the forward rate drift under the physical measure in each model is as follows:

model I/ \ d f( t ,s )

^ ' s) = t +model II

/ . x d f( t ,s )» { t ’ s ) = +

model IIId f (t s) f sH (t , s) = — — ------ 1-a ( t , s ) I a (t , a;) c (a;, s)dx + cr (£, s) I A (£, it) c (it, s) du

vs Jo Jo

In all models we assume that the forward rate risk premium A (£, s) is a function of

the spot rate and the time to maturity:

A (t , s) = r (t) A (s) (4.6)

where A (s) is a deterministic function of s.

The change of measure from the physical to the risk-neutral is given by:

rood tZ f( t ,s )= I \ ( t ,u ) c(u,s) dudt + d fZj (t,s) (4.7)

Jo

To obtain the volatility dynamics under the physical measure in model II we define

the process m (t) as follows:

dm (t) roo= A (t, y) dtZ f (t, y)dy + ip (t) dW (t)

Jom (t)

Under the physical measure, W r (t) defined by the following equation is a standard

Brownian motion satisfying assumption (I):

d W v (t) = d W ( t ) - ^ - r r d { W ( t ) , m ( t ) )Tfi ( q

= d W ( t ) - i / ) ( t ) d t (4.8)

We assume that 'if) (t) = if;, where if) is a constant. Hence, the dynamics of ln<j (t)

under the physical measure in model II are:

d In a (£) = £ [$ — In g (£)] d t + rjdW ^ (t ) (4.9)

where d = 0 + ^ . Under this specification the volatility risk premium if) is not identifiable

from forward rate data alone. In this model it would make no difference if we assumed

61

that the volatility risk premium is proportional to lncr (t): ip (t) = ip In o (t). The

dynamics of In a (t ) would be of the same form:

diner (t) = C — lntr (f)J dt + rjdWF (t)

where d = and C — C — itf*-

In model III the process rh (t) is:

d m (t) r°° r°°- ,.v = / A (t, y) dtZf (t , y ) d y + I if) (t, y) dtZa (t , y) dy

m \t) Jo Jo

Zp (t, s) defined below is a random field satisfying assumptions (I) and (II) under

the physical measure:

dtZ^(t,s) = dkZa ( t , s ) - ^ 7^ d ( Z a (t ,s ) ,m( t ) )m (tj

roo= dtZa (f, s) - / if) (t, y) C„ (s , y) dydt (4.10)

Jo

We assume that if) (t, s) = if) (s), where ip (s) is a deterministic function of the time

to maturity s. Thus the dynamics of ln<r (t, s ) under the physical measure in model III

are:

dt In o( t , s ) = £ (s) [#(s) - In o (t,s)]dt + rj (s) dtZ„ (t,s) (4.11)

where 19(s) = 0 (s ) + / 0°° ip (y) Ca (s, y) dy. As in model II, the volatility risk premium

is not identifiable from forward rate data alone.

4.3 Estimation of the models

4.3.1 M odel I

In order to estimate the models we discretize the dynamics of forward rates and

volatility according to an Euler scheme:4

'd f t .ft+1,8 f t ,S ds

r ang(s) / g (x)c (x ,s)dx + (rt) 2 g(s)R(s) Jo

A t

+g (s) VnXt+ 1,8 (4.12)

where R (s) = / 0°° A (u) c (u, s) du, X tiS ~ N (0, At) and cor (Xt,Sl, X t,S2) = c (*i, s2) .

Let T + 1 denote the total number of observations and m the total number of

maturities in the sample. A t is equal to one working day, and the distance between

different maturities in the sample is one month. We measure both t and s in terms of

We assume that forward rates are observed without error. Furthermore, we do not model any estimation errors in the derivatives '*.

62

years. Let fj denote the ra-dimensional column vector of forward rates at time i. The

likelihood function of all fj, i = 0 ,1 ,... , T, is:

= p(fo)p(fi|fo)p(f2|fi) • •••• -p (fr|fT -i) (4.13)

where p (fj) denotes the probability density function of fj. Dacunha-Castelle and Florens-

Zmirou (1986) show that for fixed At the maximum likelihood estimator of the parameters

of a discretized diffusion is consistent and asymptotically normal as T —> oo. Since the

unconditional density of fo is unknown, we maximize the approximate likelihood function,

denoted by L:

■k(fi»f2j---»fr) = p ( f i ,f2,...,fT |fo) (4-14)

The omission of p (fo) from the likelihood function is justified by the fact that this

term is dominated by the rest of the terms in the likelihood function for large T and, as

a result, does not affect results asymptotically.

Let Xf denote an m-dimensional column vector with jth element x t j equal to:

x t,j = ~ —J-— - - &sL ^ f g(x)c (x , Sj) dxAt - R (sj) rtA t (4.15)9 \sj)y /rt Jo

where Sj denotes the j th time to maturity, 1 < j < m. Let fi denote the correlation

matrix of the X tjS, with i j th element equal to c(sj, Sj). We assume the following

specification for the functions g (s), R (s), and c(si,S2 ):

g (s) = ae^s (a > 0) R (s) = 'ye5s c (si, S2 ) = e'clSl_S2l (4.16)

Model I is therefore described by the following vector of five parameters:

9 = (a (J 7 S k); (4.17)

The approximate log-likelihood function is equal to:

. T - 1 T - i / m \

^ = ~2A t ]L X 'n _ l x i - y ^ k i r i - T [ ^ ^ S j + m i n a J*=0 2=0 y j —\ J

T— — [mln(27rAt) + ln|ft|] (4.18)

The maximum likelihood estimator of 9 is given by

0 = arg max In L (4.19)0

63

4.3.2 M odel I I

The discretized forward rate dynamics in model II are:

ft+1,8 ft,8 — + of 0 (s) g(x)c (X, s) dx + atg (s) rtR (s)

+atg (s) Xt+iiS (4.20)

where X t,s ~ iV(0,At), R (s) = f £ ° \ (u )c (u , s )d u = 7eSs, and cor (Xt,Sl,Xt,S2) =

c(su s2) = e**31-52!, as in model I. Here g(s) has the following functional form:

S(») = e " (4.21)

The discretized volatility dynamics axe:

Incrt+i = In at + C [# — Inot] At + rjYt+i (4.22)

where Yt ~ N (0, At) and cor pQ1)S, Yf2) = 0 V t i , t 2,s. We rewrite the above equation

as:

In ot+i = « + f In ot + <t>Ut+\ (4.23)

where a = £ = 1 — (At, (f> — rjy/At, and Ut ~ W (0,1). If |£| < 1 then lno*

is stationary, and the unconditional distribution of In cr* is normal with mean and

variance We assume that £ satisfies the condition for stationarity. The distribution

of Inert conditional on lnot_i is normal with mean a + £ lno t-i and variance 4>2.

As before, we maximize the approximate likelihood function, since the unconditional

density of fo is unknown. The approximate likelihood function is equal to:

Z ( f 1?. . . , f T) = p ( f i , . . . , f T |f0)

= f p ( f i , . . . , fr |fo , cr)ip (<r|fo) dcr (4.24)Jrt

where <r is a T-dimensional column vector of volatilities, cr = (oq <j \ . . . <j t - i)/, and

V?(er|fo) denotes the joint probability density function of 0 0 , 0 1 , • • •, 0T-1, given fo.

Hence,

Z ( f i , . . . , f r ) = f p (fi|fojo"o) p ( 2| f i , &i) P (^31 25^2) — •p (fr lfT -iiO 'T -i) xJrt<P (cro|fo) (tfiko) • ----• (p (0T-\\0T-2) dxjodxj 1 . . . d0T- 1 (4.25)

We approximate p (oo|fo) by the unconditional density (p (ctq), as the conditional density

p (cr0 |fb) is unknown. This approximation does not affect results for large T. For

simplicity we will omit fo from density functions henceforth.

64

Let F denote an m x T-dimensional matrix with tth column f*. Equation (4.25)

implies that the approximate likelihood function L (F) can be estimated by generating A

volatility paths a and taking the average of p (F|<r), given a forward-rate sample (fo,F),

since L (F) = E [p (F|<x)]. Hence, an estimator for the approximate likelihood is

i (p ) = x I > ( EV i ) (4-26)J=1

This naive Monte Carlo estimator is inefficient, however, as A needs to be unfeasibly

large in order to obtain an accurate estimate, given a realistic sample size. There exist

a number of methods that can be employed to reduce the variance of the above Monte

Carlo estimator, one of which is importance sampling. In importance sampling volatility

samples are not generated using ip (a), but from an alternative density, the importance

sampling density ip (cr), which has the property that it is approximately proportional to

i ( ¥ ) = j p j m ^ i , HrT)da (4.27)

The importance sampling likelihood estimator, therefore, is:

1 A p ( p k j )t , F ) - A g * L ) <“ »

Geweke (1989) shows that L (F) is a consistent and, under certain conditions,

asymptotically normal estimator of L(F). The smaller the variance of , the

smaller A needs to be. When is exactly equal to a constant c, then a sample

of only one volatility trajectory is required. Since p (F|cr) cp (a) = p (F) ip (<r|F), if

ip(<r) = ip((T|F) then c = p(F) = L(F). As L(F) is the quantity which we seek to

estimate, however, this result is not helpful.

Danielsson and Richard (1993), and Richard and Zhang (2000) have developed

importance sampling techniques for the estimation of stochastic volatility models, where

the optimal importance sampling density is obtained through an iterative process.5 For

example, Liesenfeld and Richard (2003) apply the Richard and Zhang (2000) method as

follows. First a sample of volatility paths is generated from <p (cr) given a set of values

for the model parameters. Having selected a particular distribution, such as the normal

for example, for ip (at\crt-i), the parameters of this density are obtained by performing a

set of T regressions that seek to fit the denominator of the ratio in equation (4.28) to the

numerator, using the sample of volatility paths. The regression equations are constructed

5 Durbin and Koopman (1997) provide an alternative importance sampling method for models that can be approximated by linear Gaussian models.

65

by taking the logarithm of the above ratio and organizing the resulting terms in such

a way that the volatility values corresponding to a single time period appear in only

one regression equation. Due to this arrangement the parameters for ip (cTt+1 \&t) must

be known when estimating the parameters for if) {at\at-\). As a result, the parameter

calculation procedure starts from the last time period. Since the volatility sample used

to obtain the importance sampling density parameters was not generated using ip(cr),

once a set of parameters for the importance sampling density is obtained, a new volatility

sample is generated using these parameters, and the process is repeated to calculate new

parameters, until convergence is detected in these parameter values.

Convergence is not guaranteed, however, and depending on the model it may not

occur at all, as we find in our case. For this reason we propose a method for obtaining the

parameters of the importance sampling density where these parameters are consistent

with the sample used to obtain them. This is achieved by making the volatility sample a

function of the unknown parameters. By simultaneously generating the volatility sample

and calculating the parameters we eliminate the need for iterations.

We assume that ^(hiô) and V7 (lncrt | lncrt_i), 1 < t < T — 1, have the following

form respectively:

where az-, i = 0 , . . . , T —1, are the parameters to be obtained. Thus if) (In <jq) differs from

lnp(fi|ft-i,ln<rt_i) +lny>(ln<7t_i|ln<7t_2) = q_ i + ]nif) (lncrt_i| ln<7t_2) + i)t- i

where the c*, i = 0 , . . . , T — 1, are constants, and the Vi, i = 0 , . . . , T — 1, are regression

errors with mean zero.

We allow each crt to appear in two different regression equations. This is less efficient

than the global optimization procedure of Liesenfeld and Richard (2003), where at

appears in only one regression equation, but it enables us to obtain importance sampling

density parameters which are consistent with the volatility sample being used. In the

if) (ln<T*| ln<7f_i)

(4.29)

(4.30)

ip (In<To) only in the mean - ao and respectively - and ip (ln<r*| Inat-i) differs from

ip (lncrt | ln<7t_i) only in parameter a, which is replaced by at. The regression equations

used to obtain the parameters o* are as follows:

lnp(fi|f0,ln<7o) + lny?(ln<70) = cq + ln ^ (lnc70) + dq (4.31)

for 2 < t < T (4.32)

66

Liesenfeld and Richard (2003) procedure, the volatility sample cannot be generated

simultaneously with the calculation of the importance sampling density parameters,

because of the dependence of the time-t parameters on the time-(t + 1) parameters,

which implies that the regression procedure must start from the last time period. In

order to obtain a volatility sample as a function of the time-t parameters, however,

the parameters for all time periods prior to t must be known. Our construction of the

regression equations enables us to begin the calculation of the parameters from the first

time period, so that we can simultaneously generate a volatility sample as a function of

these parameters.

Let the following equation represent the regression equation for time period t:

yt,i = ct + zt,i + i't,i o < t < T — 1 (4.33)

where the subscript i denotes the ith volatility sample, zî denotes the logarithm of the

importance sampling density, and yt,i is the sum of the logarithms of p and ip. This is not

a standard non-linear regression model because both ytj and zî depend on at through

In <7* . As a result, minimizing the residual sum of squares (R S S ) is meaningless here.

Rather, we minimize R S S divided by the total sum of squares (T S S ), with R S S and

T SS defined as:A A

R S S = J ] (yM - c - zt,i)2 T S S = ] T y2u (4.34)t = l i = l

where the value of the regression equation constant is:

1 ^= (Vt,i - *t,i) (4-35)

i=1

Replacing ct by the above quantity, becomes a function of at only. The volatility

sample for each time period t is a function of at, given the volatility sample of the

previous time period if t > 0. is likely to have several local minima and, as a result,

we minimize it numerically for each t, starting from t = 0. The same set of random

numbers is used to generate the sample of volatility paths in each likelihood evaluation,

in order to ensure the efficiency of the maximum likelihood estimation procedure.6 This

set of random numbers consists of T subsets, one for each time period t. Each subset is

a sample of A numbers from a standard normal distribution and is used to obtain the

time-t volatility sample employed in the evaluation of during the numerical search

for the optimal value of the parameter at, as well as the final volatility sample for time

t used in the estimation of the likelihood function given the optimal value for at.

6See Gourieroux and Monfort (1997).

67

Let x* denote an m-dimensional column vector with j th element Xtj equal to:

rsj— at I e^xc (x , S j ) dxAt — (s^) A t (4.36)

Jof t + l , S j f t , S j

x t , j —

where sj denotes the j th time to maturity, 1 < j < m. As before, f t denotes the

correlation matrix of the X ^s-

The variables yî and zî defined earlier are given by the following equations:

zt,i =

2A t 1

2(f> 2

for 1 < t < T — 1 and

1 / r * - i i ! ~ C 2 A a Vy(M = 2 A t X° ’ Xo>i + ^ + ^ 2 ~ ^ 'cm - YZTj: J

zo,i = 1 - e2 <f>2

(lna0,i -a0)Jwhere x ^ denotes the value of the vector x* given the zth volatility sample.

Model II is described by a vector of seven parameters:

0 = (/3 j S k a £ ( f ) ) r

LetJ T - 1 T - 1 m

?< = Yi x'm° ~ 1xm lngt.< -tpYI sit=0 *=0 J=1

“ f [In |fl| + mIn(27rAt)] - ^ | - ^ln<r0,i - JZT^)

T - l1 \ x 1 — S

“ C - «)2 + -^72“ ~ ao)'2 ( f ) ^ 2 ( f )

1 T_172 ~ ^ 1x1' t - l , < “ « t ) 2

The approximate log-likehhood function is:

4.3.3 M odel III

The discretized forward rate dynamics are:

ah,f t + l , s f t , s — +ds

+ot,sXt+1,

ot,s / &t,xc (x, s) dx + ot,srtR (s) Jo

A t

(4.37)

(4.38)

(4.39)

(4.40)

(4.41)

(4.42)

(4.43)

(4.44)

68

As before, X^s ~ iV(0,A£), R(s) = A (u) c (u, s) du = 'ye6s, and

cor (XMl,X Ma) = c (s i,s2) = eKlSl-S2L

The discretized volatility dynamics are:

I n (Tt+ i,s = I n <jt ,s + Cs [# » “ 111 ° t,s] A t + » 7 * ( 4 -4 5 )

where Yâ ~ N (0, At) and cor ( lt jS, YtjU) = p (s, u). We rewrite the above dynamics as:

In<Jt+\,s = ol9 + £s Ino t,s + <f>sUt+i,3 (4.46)

where as = Cs0s&t, £s = 1 - CsAt, <f>s = ysVAt, Ut,s ~ # ( 0, 1), cor (Ut,Sl, Ut,s2) =

P (si> s2). The distribution of lucres conditional on Incrt-i,s is N (as + £s ln<7t - i )S,<^).

If K.l < 1 then ]not,s is stationary with unconditional distribution N '

We assume that satisfies the condition for stationarity for all s. In this case the

unconditional correlation between In ot,Sl and ln a^S2 is equal to:

n , , V ( i - ^ ) ( i - « 2) ..........................................cor (Inot,Si, In ot,S2) = - ----- — — --------- p(su s2) (4.47)

1 SS1SS2

We assume the following functional forms for p (x , y) , a s, £s, and 4>s:

p (x , y) = ep\x~y\ as = a0eas £s = £0e^ (j>3 = (f>0e4>s (4.48)

where p, a , ao, £, £0, <f>, and (f)0 are constants.

Let F be defined as in subsection 4.3.2, and art be an m-dimensional column vector

of volatilities with *th element equal to In o ^ Si. Furthermore, let xt denote an Tri

dimensional column vector with jth element x^j equal to

f t + 1 s- ~ f t s ~ A t f s3x tj = -----—------—------------------/ ot,xc (x , Sj) dxAt — rtR (Sj) At (4.49)O t,sj JO

where sj denotes the j th time to maturity, 1 < j < m. We approximate the integral in

the above equation by the following sum:

jrsi J 1/ ot,xc {x, S j ) dx « (Tt,Sic (s^ S j ) As - - [otiSj + crt,sxc («i, S j ) ] As

Jo 2(4.50)

where As = ^ corresponding to increments of one month.

Let Ut denote an m-dimensional column vector with j th element utj equal to

/ a 0e“®-> \ ,^ ( ^ - r ^ a r ) for 4 = 0 (451)

U(J = r ls s j ( lno'Mj -« o e °SJ -Z o e(ai ln g t-1,.,) for 1 < t < T - 1 (4.52)

In order to estimate the likelihood function we adapt the importance sampling

method used for model II. The unconditional and conditional volatility densities, ip (ctq)

and (p(at \(Tt-i), are:

y>(<r0) = I — e x po(2?r) 2 |53«nc

= -— m 1 i exp

-^ O ^ u n c U O

(27r)T |5]|2 ^- - u j E xut

respectively, where

(4.53)

(4.54)

= exp [ II /--j=l J j=l y / l and 0 = exp I (j) Sj-

j=i(4.55)

and 53, 53unc are the conditional and unconditional correlation matrices of ln<r^s

respectively.

We choose importance sampling densities ip (ctq) and ip (ert\crt-i) which differ from

(p (cto) and (p (crt\o"t-i) respectively only in parameter ao, which is replaced by ao and at

respectively. Let wt denote a vector which is equal to u* for ao = at. The unconditional

and conditional importance sampling densities axe given by the following two formulae

respectively:

Ip (<70) =(2tt) 2 |52unc|2 (p™0o

ip(at \<rt-i) = ~— m i „ exp

Bxp

[ - i w J S - 1,

■W0 (4.56)

(4.57)( 2 7 r ) T | S | 2 C 0

We obtain the parameters at, 0 < £ < T —1, by minimising the ratio | ^ | correspond

ing to the following regression equation for time period t:

yt,i = ct + zt,i + Vt,i (4.58)

where i denotes the zth volatility sample. The variables yt,i and ztj are given by:

70

for t = 0. As before, fi denotes the correlation matrix of the XtfS, and the optimal

value of the constant c* is given by equation (4.35). We use a constant set of random

numbers to generate volatility samples in order to obtain the parameters at and estimate

the likelihood function, as described in the subsection for model II.

Model III is described by a vector of ten parameters:

e = (7 S K p a0 a £0 £ <t>0 <f>)' (4.63)

LetT - 1 T —1 m1 - I l - L m r p

qi = - [ l n | f i |+ m l n ( 27rAt)]t=0 t=0 j - l

1 1 T-1 1 - 2 Uo,iS «»c"0,< - 2 z J UMS ‘u«,i + 2Wo,.S x«cw o,i

1 T_1+ 2 H WMS 1-WM (4-64)

t=lThe approximate log-likelihood function is:

lnL = In ( i ^ eq' \ (4.65)

In all three models the log-likelihood function is maximized numerically with respect

to 0 using a global optimization algorithm, since L (0) is likely to have many local

maxima.

4.4 The data

We use two datasets which consist of daily UK and US yields obtained from

Datastream. Each data point is the annual yield on a zero-coupon bond, with time

to maturity ranging from zero to 120 months, at increments of one month. We use

as many maturities as were available in order to approximate the continuous nature of

random fields as closely as possible. The data span the time period 1/4/1997-30/1/2004.

The starting date of the sample is dictated by the lack of data for long maturities at

earlier dates. We measure t and s in terms of years. Thus As = ^ and A t is equal to

the timespan of the sample (6| years) divided by the number of observations, or working

days, in the sample. For the UK the total number of observations is 1727, while for the

US it is 1732.

Denoting yields by y ( t ,s ), the following relationship holds between yields and

forward rates:

sy (£, s) = f f (t, x ) dx (4.66)Jo

71

Hence, / (t, s) is equal to the partial derivative of the above quantity with respect

to s. We calculate this derivative numerically by fitting a cubic spline to the yield

curve data multiplied by the corresponding time to maturity. We also need the partial

derivatives of forward rates with respect to s in order to estimate the models. We obtain

these derivatives numerically as well, by fitting a cubic spline to each day’s forward rate

data.

4.5 Empirical results

The maximization of the likelihood function is performed using the simulated

annealing global optimization algorithm by Press et al (2002). All numerical procedures

are implemented using the C++ programming language on a Solaris 9 UNIX workstation.

In the case of model I, which is not as computationally demanding as the other models,

we also verified the results by a grid search of 3000 points.

Tables 4.2 - 4.5 present the parameter estimates (to three significant figures) and

the maximum of the log-likelihood function for each model. Appendix F presents the

graphs of the exponential functions g (s) , R (s), c (si — S2 ) , a (s) , £ (s) , <f> (s), and

p («i — S2 ) given the estimated values of the parameters of each model.

a P 7 S K

UK

US

0.232

0.151

-0.181

-0.0513

-3.36

-6.15

0.113

-0.0411

-0.381

-1.16

Table 4.2: Model I parameter estimates

P 7 S K a £ 4>UK

US

-0.136

0.0115

-9.28

-8.30

0.218

0.379

-0.259

-1.14

-0.962

-1.36

0.318

0.380

0.402

0.337

Table 4.3: Model II parameter estimates

Tables 4.6 - 4.8 present the standard errors of the parameter estimates, calculated

numerically.

72

7 S K «o a

UK

US

-0.00291

- 8.88

-0.761

-0.559

-0.000300

-0.000655

- 0.202

-0.341

-0.236

-0.0337

P fo £ 0o 0

UK

US

-1.23

-1.029

0.585

0.0694

-0.278

-0.0762

0.369

0.571

-0.236

-0.0337

Table 4.4: Model III parameter estimates

Model I Model II Model III

UK

US

1678760

1614620

1376580

1285940

1301290

1250170

Table 4.5: Log-likelihood

a P 7 8 K

UK

US

0.000359

0.000233

0.000243

0.000257

1.87

3.80

0.065

0.119

0.00121

0.00378

Table 4.6: Model I standard errors of parameter estimates

p 7 8 K

UK

US

0.0000174

0.00000242

0.00825

0.00172

0.000192

0.0000551

0.0000257

0.000145

a £ 0

UK

US

0.000190

0.000200

0.0000290

0.0000264

0.0000486

0.0000804

Table 4.7: Model II standard errors of parameter estimates

7 8 K «o a

UK

US

0.00000302

0.0169

0.00281

0.00216

0.000000750

0.000000785

0.000572

0.000751

0.00000819

0.0000273

P £o £ 0

UK

US

0.00470

0.00129

0.00131

0.000226

0.00107

0.000223

0.00129

0.000626

0.00000777

0.0000230

Table 4.8: Model III standard errors of parameter estimates

73

¥>0 <Pi a

true values 0 0.97 0.2

mean 0.00330 0.873 0.205

st. deviation 0.0522 0.169 0.0767

skewness 3.42 -3.49 3.23

kurtosis 19.0 13.7 13.5

Table 4.9: Monte Carlo study

Table 4.9 presents a Monte Carlo study of our importance sampling technique for

estimating stochastic volatility models using Monte Carlo maximum likelihood. We

estimate the following simple stochastic volatility model:

Vt = VxiUt (4.67)

ln r t = (p0 + (pjlnxt-i + aZ t (4.68)

where Ut and Zt are independent standard normal random variables. We generate a

sample of size 1000 given the model parameter values shown in the second row of the

above table. Using the sample for yt we estimate the model 1000 times. We summarize

our results in rows three to six of the above table. The importance sampling density

differs from the actual density of \n.xt only in the mean and we obtain the optimal value

of the importance sampling density mean in the way described in section 4.3. Appendix

G presents the graphs of the distributions of the obtained estimates for parameters (p0,

(fi, and <j .

Let q denote the average of the ratio over all time periods obtained in one

likelihood evaluation for model II or III:

I F R S S '

T s > T S S t

For model II the average value of q over all likelihood evaluations performed is 0.63%

for the UK and 2.18% for the US. The standard deviations are 1.35% and 5.84%, and

the total number of likelihood evaluations are 2110 and 4000 respectively. For model III

the average value of q is 0.9% for the UK and 1.6% for the US. The standard deviations

are 6.8% and 4.9%, and the total number of likelihood evaluations are 2980 and 2200

respectively.

However, q is not a reliable measure of the fit of the importance sampling density

to the joint density of forward rates and volatility, since we allow In a* to appear in two

74

different regression equations, as explained in subsection 4.3.2. Let yj (0) denote the

value of the joint density of forward rates and volatility obtained for parameter vector

0 and the jth volatility sample given 0 , and let Zj (0) denote the corresponding value of

the importance sampling density. Denoting the total number of volatility samples used

in each likelihood evaluation by A, our estimate of the likelihood for parameter vector 0

is equal to the following mean:

A

§ * (0)

Let1 ^

c(0) = X £ (1n y j (0 ) - In Zj ( 0 ) ) (4.71)J = 1

A measure of the fit of the importance sampling density to the joint density of

forward rates and volatility is given by the following quantity:

A A

r(0) = J 2 p " W w - ^ (®) - C W 1 2 / E P“ W W ] 2 (4 -72)j = 1 j = 1

The above quantity is equal to the residual sum of squares divided by the total sum of

squares for the following model:

In yj ( 0 ) = In Zj ( 0 ) + c ( 0 ) + error (4.73)

For model II the average value of f over all likelihood evaluations is 0.19% for the

UK and 1.14% for the US. The standard deviations are 3.44% and 9.18% respectively.

However, apart from a few large outliers, most values of r (98.6% and 96.4% for the UK

and the US respectively) fall below 0.1% for A = 20. Hence, our importance sampling

method provides a highly accurate likelihood estimate. For model III the average value

of f is 0.61% for the UK and 3% for the US. The standard deviations are 6.4% and 15%

respectively, while 98.8% and 96% of the values of f fall below 0.1% for the UK and the

US respectively for A = 20.

4.6 Conclusion

Not much empirical research exists concerning multifactor term structure models.

Previous econometric studies of stochastic volatility models have concentrated on models

with a low number of latent variables, due to the estimation difficulties posed by high

dimensional models. In this chapter we estimate random field models of the term

structure of interest rates using 121 forward rate maturities and up to 121 latent variables

75

representing stochastic forward rate volatility. We use three different specifications for

the forward rate volatility function: (i) a function of the spot rate and the time to

maturity, (ii) a function of the time to maturity and a stochastic process driven by a

Brownian motion, (iii) a random field. In cases (ii) and (iii) we assume that the random

shocks driving volatility are uncorrelated with those of forward rates. The deterministic

volatility model is estimated by maximum likelihood and the two stochastic volatility

models by Monte Carlo maximum likelihood. We develop an importance sampling

technique that substantially reduces the variance of the Monte Carlo likelihood estimator

so that accurate likelihood estimates can be obtained in spite of the size of the model.

Possible avenues for future empirical research are the examination of alternative

specifications for the random field models, such as the incorporation of jumps in forward

rates and/or volatility, and allowing for correlation between the random field driving

forward rates and innovations in volatility; the comparison of random field models with

other types of term structure models; the use of option price data to test the models

and estimate the volatility risk premium; and the estimation and prediction of volatility.

Moreover, there is a need for further research on econometric methods for estimating

high-dimensional stochastic volatility models.7

7 An implementation of MCMC was found to require more computer time than Monte Carlo maximum likelihood estimation, while it exhibited very poor convergence (slow mixing) in the estimated parameter and volatility values. Even in the case of model I, which is the simplest, MCMC takes much longer than maximum likelihood estimation. Due to the complexity of the model, Metropolis-Hastings algorithms need to be used for sampling at least three of the five parameters of model I. This implies that the likelihood function needs to be evaluated more times in MCMC than in maximum likelihood, in order to obtain a large enough sample of parameter values. In the MCMC estimation of models II and III individual likelihood evaluations are as fast as in model I, but the number of simulations has to be much larger due to the inclusion of volatility in the estimated parameters.

76

Chapter 5

Conclusion

In this thesis we have considered random field models for the implied volatility

of stock options and the term structure of interest rates. Random field models offer

greater flexibility in fitting observed asset prices than previously studied finite-factor

models. This advantage can eliminate the need for periodic recalibration of the model

as new market data become available. It also allows greater precision in modelling the

correlation between assets with similar characteristics, which can be very valuable in

forming hedging portfolios for exotic derivatives using standard options or bonds.

In general, finite-factor models with n factors predict that any derivative asset can be

perfectly hedged using any n bonds in term structure models, or n plain-vanilla European

options in stochastic volatility stock price models, whereas in practice derivatives tend

to be hedged with assets of similar maturity. While in principle, the entire continuum of

bonds or options, depending on the model, is required to hedge a derivative instrument

in random field models, it may be possible to attain a near-perfect hedge using only a few

assets, if those assets are strongly correlated with the derivative. In the case of interest

rate contingent claims it is usually easy to select those bonds which are likely to be highly

correlated with the derivative instrument. Selecting options to hedge other derivatives

may be less easy, however, as options have an additional exercise price dimension. In

this case, the appropriate options may be chosen on the basis of liquidity issues.

Although random field models are infinite-factor models, it is still possible to obtain

closed-form solutions for the prices of derivative assets, as we do in chapter 2 for variance

swaptions and in chapter 3 for standard interest rate contingent claims. Furthermore,

random field models can be estimated using a small number of parameters. In chapter

4 we estimate three random field models of the term structure of interest rates, with

deterministic and stochastic forward rate volatility. We use 121 forward rate maturities

in order to approximate as closely as possible the continuous nature of random field

models using available data. Previous empirical studies of random field models in finance

use a very small number of factors in the estimation process, which makes the estimated

models indistinguishable from finite-factor ones. This reduction of random field models

to finite-factor versions was considered necessary in order to make the estimation problem

computationally feasible. By developing an appropriate importance sampling method, we

are able to reduce the computational demands of the problem and estimate the stochastic

volatility models using Monte Carlo maximum likelihood. There exists substantial scope

for further theoretical and empirical research in the area of random field asset pricing

models.

78

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88

A ppendices

A. The Ito-Venttsel formula

Let W i ( t) , i = 0 ,1 ,... , n, be independent standard Brownian motions, and / (t, u)

a stochastic process with the following dynamics when u is not stochastic:n

df (t , u) = a (t , u) dt + Pi (£, u) dW* (t)z=0

If v (£) is a stochastic process with dynamics

n

dv (t) = 7 (t, u) dt + (£, v) dWi (t )*=o

then / [t, v (£)] satisfies the following stochastic partial differential equation:

df [t, v (t)] = a[t,v (t)] dt + ^ 2 f t [*»v (01 W + Wi= 0

+l ^ f l 5i^V'>dt + Y ,^ S t’ dt (A-!)z=0 z'=0

(See Brace et al. (2001) and references therein.)

B. Proof of lemma 2.3

Equation (2.36) is obtained by a simple extension of Shephard’s (1991) proof for

formula (2.33). Let G (y) = sign (y), y G [—h, h\. The Fourier transform of G(y) is

$ (*>; h) = f H G (y) e '^d y = 2 [cos( M ~ l ]J-h W

The convolution of G (y) with the product p{y)eay, where p (y) is the continuous

probability density function of y, is equal to:

^ ( K \ h ) = f p (y) eayG (K — y ) dy= f p(y)eayd y - f p(y)eaydy J-h J-h Jk

89

where h > K and h > 0. The characteristic function of y is:

/ooe%(pyp(y)dy

-o o

The Fourier transform of \I/ (if; h) is equal to:

2 [cos (hep) - 1] f ° ° i w + a v ^ t - . \ j . . 2 tcos ( M ~ !]r ei(py+ayp (y) dy = 2 [cos( M ^ f (ip _ ia)J - OO *<PW J —oo

where we use the fact that the Fourier transform of a convolution of two functions is

equal to the product of the Fourier transforms of the two functions. By inverting the

Fourier transform of (if; h) we obtain:

lim ( if ; h) = E (eQy) - 2E (eayIy>K)h —t o a

= A l t o27T /l->00 J_00 up

7T Jq h->oo

_ 2 r°° r/ ( - *«)e_i W o

up

dip

dip

up

Combining this with the relationship E(eay) = / ( —ia), we obtain the required

result. □

C. Proof of proposition 3.1

(Goldstein, 2000) Let U(t,s) = — f* f (t,x) dx. Then P{t,s) = eu(t,s). The

dynamics of U (t, s) are:

dtU (t,s) - r ( t ) d t - J a (£, y) dydt — o (t,y) dtZ (t,y) dy

Using Ito’s lemma,

= dtU(t,s) + \ d t(U(t , s) )

= r (t) d t — f a (t, y) dydt — f a( t , y )dtZ [t,y)dy

+ \ I I a (t , y) a (i, x) c (rr, y) dydxdt

Under the risk-neutral measure the drift of bond prices must be equal to the spot

rate:dtP ( t , s)

( t ) dt — J a ( i , x) dtZ ( t , x) dxP (t,s )

Therefore, the following condition must hold for no arbitrage opportunities to exist:

J a ( t , y ) d y = ^ J J a(t ,y)cr(t ,x)c(x,y)dydx

By differentiating the above equation with respect to s we obtain the required result.

□

D. Proof of proposition 3.2

(Goldstein, 2000) Suppose that V (t) represents the value at time t of a portfolio

consisting of 77 shares of the bank account B (t) and 6U shares of the zero-coupon risk-free

bond maturing at time u, V u > t:

/oo6UP (£, u) du

Hence,

/ooOudtP (t, u) du

Under the risk-neutral measure Q, is a martingale. Under the T-forward

measure = pftj') is a martingale. Using Ito’s lemma we obtain the dynamics

of VT {t):

= a { t ’ x ) d x + J t a ^ dudx

x d t Z ( t , x ) + £ <j(t,y)c(x,y)dyd1^

The above equation implies that the T-forward measure is given by equation (3.10).

□

E. The forward-rate drift under the risk-neutral measure

in section 4.2

(Santa-Clara and Sornette, 2001) The following relationship holds between bond

prices and forward rates:ru—t

In P (t,u) = — I f (t , x) dxJo

From the above equation it follows that:ru—t

dt InP (£, u) = f (t,u — t) dt — I dt f ( t , x) dxJo

Using Ito’s lemma we obtain the dynamics of bond prices:

dtp(t'u) = \^t,u~t + \ L L a(t’x)a(t’y)c(x’y)dxdv

(E.1)

dt

91

Under the risk-neutral measure the drift of bond prices must be equal to the spot

rate. Therefore,

ru—t j r-u—t ru—t/ a(t ,x) dx = f (t,u — t ) + - / a (t,x) o (t,y) c(x,y) dxdy — r (£)

Jo * Jo Jo

Let s — u — t. Taking the derivative with respect to u in the above equation we

obtain:

a (£, s) = d-f s) +cr(t,s) f a (t, x) c (s, x) dx □ (E.2)os J0

92

F. Graphs of exponential functions with estimated parameters in models I, II, and III

0.200.150.100.05

Figure 2. Model I US g(s) function

0.30.20.10.0

Figure 1. Model I UK g(s) function

Figure 4. Model I US R(s) function

-5-10-15

Figure 3. Model I UK R(s) function

0.5

0.0

Figure 6. Model I US c(si~S2) function

0.5

0.0

Figure 5. Model I UK c(sr S2) function

1.151.101.05-1.00

0.5

0.0

Figure 7. Model II UK g(s) function Figure 8. Model II US g(s) function

93

-50

-100

Figure 9. Model II UK R(s) function

-200

^00

Figure 10. Model II US R(s) function

0.5

0.0

Figure 11. Model II UK c(si-sz) functionFigure 11. Model II UK cfsj-s^) function

0.5

0.0

(0 000

D 20 40 60 80 100 120- ■ | —|------ 1------ 1

-0.001-0.002-0.003

Figure 13. Model III UKi?fo) function

-5

Figure 14. Model III US R(s) function

1.0000.9990.9980.997

1.000

0.995

0.990

Figure 15. Model III UK c(sr sz} function Figure 16. Model III US c(sr S2) function

94

- 0.20-0.25-0.30-0.35

Figure 18. Model III US a(s) function

0.0- 0.1- 0.2-0.3

Figure 17. Model III UK a(s) function

0.080.060.040.02

Figure 20. Model III US %(s) function

0.60.40.20.0

Figure 19. Model III UK %(s) function

0.6

0.5

0.4

Figure 22. Model III US <f)(s) function

0.4

0.2

0.0

Figure 21. Model III UK <f)(s) function

0.5

0.0

0.5

0.0

Figure 23. Model III UK p(sr si) function Figure 24. Model III US p(sr S2) function

95

G. Monte Carlo study results

20

0 4-0.3 -0.2 -0.1 0.0 0.2 0.3 0.5

Figure 25. Distribution of estimated values for (po

-0.4 - 0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2

Figure 26. Distribution of estimated values for (p}

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Figure 27. Distribution of estimated values for cr

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